ofloy 126 Stat 571 Probability \& Messure Lecture 5. Measure Furetions Simple furtion (simple rendom veriable) net $(\Omega, \mp, \mathbb{P})$ be a probability space be $\mathbb{P}(\Omega)=1$ Simple revetom veriable $x: \Omega \rightarrow \mathbb{R}$

L $x(w)$ only takes a finite \# of values $\left\{x_{1}, x_{p}\right\}$ also, the set $\left\{w \in \Omega: x(w)=x_{i}\right\} \in \Psi$ Let $\left[A_{i}\right]_{i=1}^{P}, A_{i} \in F$ be a dirjoint portitus of $\Omega$, $L_{i=1}^{p} A_{i}=\Omega \quad A_{i} \cap A_{j}=\phi \quad$ f $\quad c \neq j x(\omega)=\sum_{i}^{p} x_{1}\left[\omega \in A_{i}\right]$ We can then gey that the probability that $x$ is epual to $x_{c}$ is $\mathbb{P}\left(x=x_{i}\right)=\mathbb{P}\left(\left\{\omega \in \Omega: x(\omega)=x_{i}\right\}\right)=\mathbb{P}\left(A_{i}\right)$ can also deline the expectation of $x$ to be
$$E x=\left\{x_{i} \mathbb{P}\left(x=x_{i}\right)\right.$$

Example: $\Omega=(0,1]$
$x_{1}=\overline{4}$

\begin{figure}
\includegraphics[alt={},max width=\textwidth]{https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-1.jpg?height=643&width=800&top_left_y=2445&top_left_x=1269}
\captionsetup{labelformat=empty}
\caption{Simple measurathe function $(\Omega, F, \mu)$ be a meaxue space $F: \Omega \rightarrow R$ is st $F(\omega)=\sum_{i=1}^{p} x_{i}$ II $\left[\omega \in B_{i}\right]$}
\end{figure}
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-1.jpg?height=1819&width=2352&top_left_y=3714&top_left_x=0) So are $f+g, f \cdot g=f(\omega) g(\omega)$, mex $\{f, g\}$ min $\{f, g\}$
are also simple funtions
We define the integral of a simple finetion to be
$$f f d_{\mu}=\sum_{i x, \mu(B i)}^{a}$$

![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-1.jpg?height=3058&width=2367&top_left_y=4576&top_left_x=0)
$$\begin{aligned}
& \text { Secordly, }(Y(, Y) \text { is otten }(\mathbb{R}, B(\mathbb{R})) \\
& \text { ( } \left.\mathbb{R}, \beta(\mathbb{R})^{+}\right) \\
& \text {In this ase, me a say that } f \text { is Borel measurable } \\
& \text { Similarly, me can replace } \beta(\mathbb{R}) \text { with } M_{\lambda}(\mathbb{R}) \text { to } \\
& \text { Set lebesgue measurable finctions. }
\end{aligned}$$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-1.jpg?height=157&width=1301&top_left_y=8902&top_left_x=62&polygon=1270,0,1301,32,1301,126,1285,142,533,157,361,157,0,142,0,32,126,16,1223,0)

\section*{i.e. for images of set firections preserve set perchous}
$$\begin{aligned}
& \text { Anint : If I wait to show thet sets A,B are } \\
& \text { equal, thein shal thet both } A \subseteq B \text { and } B \in A \text { hold })
\end{aligned}$$
g. het $A_{t}=(-\infty, t] \forall t \in \mathbb{R}$ these sets generte $p(R)$
$\Rightarrow f$ is measurable as long as $\{x: f(x) \leqslant t\}$ are
measu-ble
2. For any $A \in X$, the malicator function $f(x)=\mathbb{I}[x \in A$ is measurester, The $\theta$-field generented by $f^{-1}$ is simply
$$\text { 3. For reasurcule knotions } f, g \cdot X \rightarrow \mathbb{R} \text { then } f+g$$
following ane also measureble
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-1.jpg?height=267&width=1975&top_left_y=12131&top_left_x=78&polygon=1943,0,1959,16,1975,47,1975,173,1959,188,1317,235,564,267,16,267,0,220,0,141,16,126,78,94,282,63,1379,0)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-1.jpg?height=236&width=2348&top_left_y=12570&top_left_x=31) a courtable pacrsection of messivelle sits is restivelse
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-1.jpg?height=189&width=2285&top_left_y=12931&top_left_x=94) $=$ " $\left.\left\{x: f^{(x)}\right\}+1\right\}$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-1.jpg?height=205&width=1129&top_left_y=13448&top_left_x=78)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-1.jpg?height=266&width=1156&top_left_y=13655&top_left_x=87)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-1.jpg?height=189&width=2227&top_left_y=13903&top_left_x=125)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-1.jpg?height=204&width=2335&top_left_y=14075&top_left_x=31&polygon=2288,0,2320,31,2335,94,2335,125,2320,188,2304,204,31,204,0,172,0,78,16,63,141,31,2273,0)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-1.jpg?height=205&width=2274&top_left_y=14294&top_left_x=31)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-1.jpg?height=220&width=2273&top_left_y=14451&top_left_x=31&polygon=2226,0,2257,16,2273,31,2273,157,2241,173,439,220,63,220,31,204,0,173,0,79,16,63,501,31,1411,0) over
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-1.jpg?height=330&width=2336&top_left_y=15047&top_left_x=31&polygon=2289,0,2273,16,2242,16,2226,32,1411,32,1395,47,1348,47,1333,32,1286,32,1270,47,1176,47,1160,63,1129,63,1113,79,1035,79,1019,94,1003,94,988,110,972,110,956,94,925,94,909,110,878,110,862,126,47,126,32,141,16,141,0,157,0,298,16,314,32,314,47,329,157,329,173,314,267,314,282,298,361,298,377,314,392,314,408,298,424,298,439,282,518,282,533,267,549,267,565,282,596,282,612,267,894,267,909,251,1098,251,1113,235,1395,235,1411,220,1521,220,1536,204,1960,204,1975,188,2179,188,2195,173,2257,173,2273,188,2304,188,2320,204,2336,188,2336,32,2320,16,2304,16)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-1.jpg?height=283&width=2321&top_left_y=15235&top_left_x=31&polygon=2054,0,2022,32,1913,32,1897,47,1521,47,1505,63,1286,63,1270,47,1223,47,1207,63,486,63,471,79,455,63,377,63,361,79,94,79,79,94,16,94,0,110,0,204,16,220,16,267,110,267,126,282,173,282,188,267,204,282,267,282,282,267,314,267,314,251,298,235,298,220,314,204,330,204,345,188,392,188,408,204,455,204,471,188,486,188,502,204,580,204,596,188,612,188,627,204,674,204,690,188,768,188,784,204,815,204,831,188,894,188,909,204,925,204,941,188,972,188,988,173,1003,188,1098,188,1113,173,1129,188,1176,188,1192,204,1207,188,1270,188,1286,204,1317,204,1333,188,1364,188,1380,173,2289,173,2320,141,2320,47,2304,32,2304,16,2273,16,2257,32,2195,32,2179,16,2132,16,2116,0) West wer $\sigma\left(\Sigma f^{-1}(\omega): u \subset R\right.$, ven $\left.\xi\right) \leq \beta(x)$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-1.jpg?height=235&width=2147&top_left_y=15580&top_left_x=31&polygon=2132,0,2147,15,2147,125,2132,141,2085,156,157,235,78,235,31,219,0,203,0,62,31,47,313,31,1034,0)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-1.jpg?height=266&width=2132&top_left_y=15674&top_left_x=31&polygon=2100,0,2116,15,2132,62,2132,141,2116,188,2100,203,721,266,392,266,47,251,31,235,0,172,0,125,31,94,1520,0)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-1.jpg?height=235&width=2317&top_left_y=15862&top_left_x=62&polygon=2304,0,2317,31,2317,78,2317,94,2317,141,2304,157,2038,172,1427,204,126,235,16,235,0,219,0,31,1709,0)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-1.jpg?height=249&width=2171&top_left_y=16011&top_left_x=59) vi,vety
Let $(\Omega, 7, \mu)$ Lo a maxtro speci for $f, g: e \rightarrow \mathbb{R}$, and ong that $f=9$ a.e When the Beet $N=\left\{N: f(u) \neq g^{(L)}\right\} \mu(N)=0$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-1.jpg?height=267&width=2007&top_left_y=17367&top_left_x=141&polygon=0,0,0,157,16,172,16,219,31,235,31,251,47,266,63,251,63,219,110,172,549,172,564,188,596,188,611,172,690,172,705,188,799,188,815,204,846,204,862,188,1160,188,1176,172,1662,172,1677,157,1818,157,1834,141,1850,141,1865,157,1928,157,1944,141,2006,141,2006,0) for awth probablify 7
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-1.jpg?height=205&width=2242&top_left_y=17711&top_left_x=31) excepplee Let $((0,1,3, \beta, 3)$ be a meave speev. We $f(t)=0 \quad \forall t \in(0,1]$. Let $g(t)= \begin{cases}0 & \forall t<c(0,1] \backslash Q \\ 1 & \forall t<0,1] \cap Q\end{cases}$ Cham: $f=y$ a.e.
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-1.jpg?height=424&width=550&top_left_y=18605&top_left_x=1786)
that is, $\lambda\left(\mathrm{C}_{0}, 179 \mathrm{Q}\right)=0$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-1.jpg?height=314&width=2211&top_left_y=19326&top_left_x=94)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-1.jpg?height=221&width=2211&top_left_y=19545&top_left_x=62) $m_{1}$ and notert $\left(\tau_{m}=\Sigma 2^{-m}, q_{m}+\Sigma 2^{-m+1}\right)$

Lecture 6: Infegertion \& Gonvergerve Theorens
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-2.jpg?height=247&width=1482&top_left_y=0&top_left_x=403)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-2.jpg?height=247&width=1728&top_left_y=246&top_left_x=358&polygon=1683,0,1728,45,1728,157,1706,180,1257,225,718,247,113,247,45,225,0,180,0,112,23,23,853,0)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-2.jpg?height=293&width=1707&top_left_y=448&top_left_x=358)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-2.jpg?height=270&width=1415&top_left_y=785&top_left_x=358)
Notation. I 1
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-2.jpg?height=405&width=1751&top_left_y=1121&top_left_x=314)
$\_\_\_\_$

\section*{![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-2.jpg?height=696&width=1998&top_left_y=2109&top_left_x=44) $\therefore A \subset \mathcal{C} \Rightarrow I=$ T
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-2.jpg?height=426&width=2266&top_left_y=2961&top_left_x=44&polygon=1750,0,2087,45,2244,90,2266,135,2266,269,2244,337,1863,359,337,426,23,426,0,404,0,112,45,90,1459,0)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-2.jpg?height=360&width=853&top_left_y=3410&top_left_x=269) $\therefore 6$ or $+9=$ \\ ![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-2.jpg?height=292&width=1190&top_left_y=4173&top_left_x=0)}
$f_{(\omega)}^{+}=\left\{\begin{array}{cc}f(\omega) & \text { if } f(\omega) \geqslant 0 \\ 0 & \text { elsols }\end{array}\right. f^{-}(\omega)=\sum_{\text {else }}-f(\omega)$ if $f(\omega) \leq 0$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-2.jpg?height=10749&width=2379&top_left_y=5339&top_left_x=0)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-2.jpg?height=2491&width=2379&top_left_y=16289&top_left_x=0)
and $g^{*} \uparrow f$ so one again $\rho_{g^{*}} d_{\mu} \uparrow \int f d_{k}$ out $g^{*} \leq f_{i}$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-2.jpg?height=338&width=2245&top_left_y=19654&top_left_x=44)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-2.jpg?height=203&width=1864&top_left_y=19968&top_left_x=44)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-2.jpg?height=292&width=944&top_left_y=20148&top_left_x=44)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-2.jpg?height=338&width=1302&top_left_y=20372&top_left_x=67) ∴ form and
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-2.jpg?height=315&width=1370&top_left_y=20686&top_left_x=201)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-2.jpg?height=270&width=1549&top_left_y=20888&top_left_x=89) b. a good $f=f^{+} f^{+}$, we have $f^{+} \uparrow f^{+}$and folf
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-2.jpg?height=270&width=1100&top_left_y=21404&top_left_x=67) ∴ From concre $\int f_{i}^{-} d_{k} d \int f^{-} d x \because \quad \sin f_{i} d_{\mu} \upharpoonleft \int f d_{\mu}$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-2.jpg?height=315&width=1258&top_left_y=22302&top_left_x=44)
Theorem (Fifery leammal):
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-2.jpg?height=292&width=2042&top_left_y=22885&top_left_x=67&polygon=1997,0,2019,22,2042,157,2042,202,2019,247,269,292,67,292,22,247,0,157,0,67,22,45,179,22,404,0) Then, $\rho_{\text {limuin }} f_{i} d_{\mu} \leq$ limini $\int f_{i} d \mu$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-2.jpg?height=293&width=1550&top_left_y=23603&top_left_x=44)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-2.jpg?height=360&width=1549&top_left_y=23917&top_left_x=89&polygon=1347,1,1325,23,427,23,405,45,1,45,1,270,23,292,23,337,46,360,68,337,68,270,90,247,292,247,315,225,764,225,786,202,876,202,898,225,966,225,988,202,1526,202,1549,180,1549,45,1526,23,1504,23,1482,1) Then s $_{j} \uparrow$ miniff and $f_{i} \geq 0$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-2.jpg?height=383&width=1975&top_left_y=24388&top_left_x=112&polygon=1750,1,1728,23,45,23,23,46,23,135,0,158,0,225,23,248,23,270,0,292,0,337,45,382,67,360,696,360,718,337,785,337,763,315,808,270,853,270,875,248,920,248,943,270,987,270,1010,248,1077,248,1100,225,1122,225,1144,248,1212,248,1234,225,1952,225,1975,203,1975,23,1952,23,1930,1,1862,1,1840,23,1773,23) $\therefore \quad \int_{j_{j}} d_{\mu} \leq \int f_{i} \alpha_{\mu} \theta_{i}$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-2.jpg?height=695&width=2311&top_left_y=25129&top_left_x=44&polygon=2289,0,2311,22,2311,471,2289,493,2222,516,808,650,539,673,247,695,225,695,135,650,45,516,23,471,0,359,45,22,68,0)
Theorem (conoputed amergenac)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-2.jpg?height=360&width=2178&top_left_y=25936&top_left_x=44&polygon=1347,1,1325,23,1302,23,1302,46,1280,68,1257,68,1235,91,517,91,494,113,315,113,293,135,46,135,46,180,23,203,23,248,1,270,23,293,23,315,68,315,91,337,158,337,180,360,248,360,270,337,360,337,382,315,405,315,427,293,2177,293,2177,91,1886,91,1863,68,1841,68,1818,46,1796,46,1773,68,1729,68,1684,23,1639,23,1594,68,1571,68,1504,1,1459,1,1437,23,1392,23,1370,1) If $\left|f_{i}\right| \leq g \quad \forall_{i}$ and $f(b) \rightarrow f(c)$ to ench $w \in \Omega$ then
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-2.jpg?height=248&width=1998&top_left_y=26654&top_left_x=44) Then $f_{1}^{1} \leq f_{i} \leq f_{i}^{2} \quad$ Not them $\int f_{i}^{\wedge} d \mu \uparrow \int f_{d p}$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-2.jpg?height=336&width=2221&top_left_y=28001&top_left_x=89&polygon=2221,0,2221,157,2199,201,943,314,427,336,157,336,0,291,0,44,23,22,292,0) $f_{1}^{\wedge} d_{\mu} \leq \int f_{1} d \mu \leq \int f_{1}{ }^{v} d \mu$
$\int_{f f_{\mu} \leq} \leq \int f_{\mu} \leq \int f d_{\mu} \leq$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-3.jpg?height=238&width=1664&top_left_y=0&top_left_x=438)

\begin{tabular}{|l|}
\hline Halfusay Point, May one more lecture on Integretion. Lespales, lav of longe mumbers, CLT Jetc imoming use mersothe fortions to define a measure → bettrem two measonse spenes \\
\hline - Let $\psi: X \rightarrow Y$ be resomble \\
\hline - Let $\mu$ be a measure $X$ then we can desithe
$$v=\mu \cdot \psi-1 \text { " a megone on } \mathcal{Z}_{s}^{\prime \prime} \text { the } v \text {-fillal }$$ \\
\hline For some $\beta \in \mathcal{L}, \mathcal{U}(\beta)=\mu\left(\psi^{-1}(\beta)\right)$ \\
\hline \\
\hline Theorem: Let $F: \mathbb{R} \rightarrow \mathbb{R}$ that is ron-constant right continuives avel rov-dearcaity. Then, $\exists$ a unique measure $d F$ on $\mathbb{R}$ s.t $\forall a, b \in \mathbb{R}$ with $a<b \quad \alpha F((a, b j)=F(b)-F(a)$ \\
\hline Proof: Let $F(\infty)=\lim _{x \rightarrow \infty} F(x)$ and $F(-\infty)=\lim _{x \rightarrow-\infty} F(x)$ \\
\hline Define an open interval $I=(F(-\infty), F(\infty))$ Define $g(y)=\inf \{x \in \mathbb{R} \quad y \leq f(x)\}$ \\
\hline \\
\hline \begin{tabular}{l}
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-3.jpg?height=148\&width=162\&top_left_y=3515\&top_left_x=65) \\
Stow it makes serse claim: $g$ is left continuous and non-decreasing and for $y \in I$, $\lambda \in \mathbb{R}, g(y) \leq x$ iff $y \leq F(x)$
\end{tabular} \\
\hline Fix a $y \in I$ and let $J_{y}=\{x \in \mathbb{R}: y \leq F(x)\}$ as $F$ is non-decreasing if $x \in J_{y}$ and $x^{\prime} \geq x$ then $x^{\prime} \in J_{y}$ (cses ron-decoent) hit) \\
\hline as $F$ is right-continuous if $x_{n} \in F_{y}$ and $x_{n}$ then $x \in F_{y}$
$$\therefore J_{y}=[g(y), \infty)$$ \\
\hline Forthemore, $g(y) \leqslant x$ iff $y \leq f(x)$ Also for $y \leq y^{\prime}$ we have $J_{y}^{\prime} \subseteq J_{y}$ \\
\hline $\therefore g(y) \leq g\left(y^{\prime}\right)$ So if $y_{n} \uparrow y$ \\
\hline Then $J_{y}=\widehat{n}_{n=1} J_{y_{n}}$ and this $g\left(y_{n}\right) \rightarrow g\left(y_{n}\right)$ \\
\hline \begin{tabular}{l}
rowerdring \\
$\therefore 9$ is cont from the left
\end{tabular} \\
\hline Next, $g$ is borel resurable! "invesse rajos produce ∴ Defining $d F=\lambda \cdot g-1$ gives $d F((a, b])=\lambda(\{y: g(y)>a, g(y) \leq b\})$
$$=\lambda((F(a), F(b)])$$ \\
\hline any other measure $\mu$ s.t $\mu((-b])=F(b)-F(a)$ must comender with $d F$ on all bovel sets. \\
\hline
\end{tabular}

\section*{Def: (Ladon measure) \\ Let $(\Omega, \beta, \mu)$ be a reasure spore $\beta$ s the Borel $\sigma$-ficld
$\mu$ is sall to be knolon if $\mu(k)$. Inite for all compact $k \epsilon \beta$}
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-3.jpg?height=238&width=2194&top_left_y=8626&top_left_x=18)

\begin{tabular}{|l|}
\hline \begin{tabular}{l}
$\therefore F(b)-F(a)=\mu((a, b])$ for $a<b$ \\
$\therefore \mu=d F$ by uniqueness
\end{tabular} \\
\hline \begin{tabular}{l}
Product preasure (tale ${ }^{2}$ ) writy the integel sproduct signm fied" \\
aiven $x, y, y$-fields me define $x \times y$ to be the O-fivel geverted by the rectaples $A \times B$ where $A \in X$, $B \in \mathcal{U}$. We denote the set of all restangles to be $R$, $A$ signas aread $\lambda \sigma(p)=: x \times y$ \\
signn field what \& $x$ ?
\end{tabular} \\
\hline \\
\hline \\
\hline \begin{tabular}{l}
Theorem (Existence + Uniqueness of Product measure) Let $(X, X, \mu)$ and $(Y, Y, v)$ be $\delta$-finite measure spacies. Let $\pi$ be a set function on $x \times y$ s.t for $A \in X$ and bey $\pi(A \times B)=\mu(A) \nu(B)-$ Stert Then, $\pi$ extends uniquely to a measure on $(X \times Y / X \times Y)$ S.x. for any $E \in X \times Y \quad \pi(E)=\iint \mathbb{1}_{E}(x, y) d \mu d \nu =\iint \mathbb{1}_{E}(x, y) d \nu(y) d \mu(x)$ \\
Proof: can't axtend to general \\
(the extervion won't be inique) \\
(1) Do it for finite measures \\
(2) Extenal to $v$-finite
\end{tabular} \\
\hline Falt 1: On $R$, set firition $\pi$ is countarbly odditive we can extend $R$ to a field $A$ by including countable semi-ning unions of Rectagles \\
\hline We need a theorem smilar in style to Dynuin $\pi-\lambda$ Def: (Monotone closs) </ similor to Dyphin-Pi Theorem A collection $M$ of subsets of $\Omega$ is monotone if (1) for $\left\{A_{i}\right\}_{i=1}^{\infty}$ s.t. $A_{i} \in M$ and $A_{i} \uparrow A=\bigcup^{\infty} A_{i}$ then $A \in M$ "closed inder irrienty segvents it sets" (2.) for $\left\{A_{i}\right\}_{i}^{\infty}, A_{i} \in M$ and $A_{i} \downarrow A=\bigcap_{i}^{\infty} A_{i}$ then $A \in M$ \\
\hline Note: if a field $A$ is also Monotone then it is a $J$-field Recall $A$ is a field if $-\phi, \Omega \in A$ - if $B, A \in X$ then $B \mid A \in A \left\{\begin{array}{l}\text { - if } B, A \in A \text { then } B \cup A \in A \\ \text { replace: } A \in A \Rightarrow A^{c} \in A \text { inte }\end{array}\right.$ \\
\hline
\end{tabular}

Theorem chonotine class Theoren
Let $A$ be a field a m be a monstane closes s.t. $A<m$, then $f(A) \leq M$.
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-3.jpg?height=7142&width=2379&top_left_y=16246&top_left_x=0)
and $\iint \Sigma_{\varepsilon}\left(x_{1}\right) d \alpha_{1}(\mathcal{G}) d \mu A \in F$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-4.jpg?height=379&width=1078&top_left_y=7076&top_left_x=757)

Proof C For ve
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-4.jpg?height=351&width=1603&top_left_y=7862&top_left_x=378) $T(\varepsilon)=\iint_{E}(x, y) d_{\mu}(x) d x(y)$ any $\varepsilon_{0} \cdot x \times y$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-4.jpg?height=438&width=1923&top_left_y=8561&top_left_x=58)

Whomsh: st. $\rho(A \times B)=\mu(A) \gamma(B)$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-4.jpg?height=380&width=1661&top_left_y=9755&top_left_x=87)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-4.jpg?height=351&width=1399&top_left_y=10250&top_left_x=58)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-4.jpg?height=409&width=1166&top_left_y=10774&top_left_x=87)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-4.jpg?height=613&width=1078&top_left_y=10716&top_left_x=1223) -, $\mu\left(N_{i}\right)<\infty \quad v\left(B_{i}\right)<\infty$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-4.jpg?height=350&width=991&top_left_y=11881&top_left_x=87)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-4.jpg?height=467&width=962&top_left_y=12259&top_left_x=145)
$(z)=\iint p_{E}(x, y) \alpha_{\mu}(x) d x(y)$ Fordrowe,
wedine,
$\therefore \pi$ a metare convery For any othe antosle wedit Rivity + mper (8) soum for
$\pi(\epsilon)=\sum_{(j)} \pi\left(\epsilon_{j}\right)=\sum_{j,} \rho\left(\epsilon_{i j}\right)=\rho(c) \square$ numering ut $c$ Then $\nu=\| \iint f(x, y) d x(y) d \mu(x)$
Not $\int f(x, y) d \mu(x) y$-morombe

Prover have his? for modictor forstoms maducter firstions
$\_\_\_\_$
$\_\_\_\_$ smoke finterns
(ff) $d(\mu \times \nu)$ <

\begin{figure}
\captionsetup{labelformat=empty}
\caption{mide 6}
\includegraphics[alt={},max width=\textwidth]{https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-4.jpg?height=321&width=350&top_left_y=17356&top_left_x=1485}
\end{figure}
ol f-

\begin{figure}
\captionsetup{labelformat=empty}
\caption{mide 6}
\includegraphics[alt={},max width=\textwidth]{https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-4.jpg?height=380&width=321&top_left_y=17297&top_left_x=2009}
\end{figure}
$\quad \int f^{+}(x, y) d \mu^{(x)}<\infty$
$\quad f^{-}(x, y) d y(y)<\infty \quad \mu$.
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-4.jpg?height=437&width=2098&top_left_y=17822&top_left_x=174) $\int f(x, y) d v(y)=\int f^{+}(x, y) d v(y)-\int f(x, y) d v(y), \ldots, c \iint f(x, y) d v(y) d \mu(x)=\iint f^{+}(x, y) d v(y) d \mu(x)$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-4.jpg?height=584&width=2127&top_left_y=18899&top_left_x=58&polygon=728,1,699,30,29,30,29,59,0,88,0,350,29,379,29,408,0,437,0,554,29,583,117,583,175,525,233,525,262,554,291,554,291,496,321,467,321,437,350,408,495,408,524,379,641,379,670,408,787,408,816,379,845,379,874,408,1136,408,1165,437,1194,408,1922,408,1951,379,2068,379,2126,321,2126,146,2068,88,2068,59,2039,59,2010,30,1398,30,1369,1,1340,1,1311,30,1282,1,1136,1,1107,30,1019,30,990,1,961,30,787,30,757,1)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-4.jpg?height=438&width=1486&top_left_y=19540&top_left_x=29) spaces Then, $\Omega=\Omega_{n}$
$w_{i=1}^{\infty} \Omega_{i=1}^{\infty}$ s.t. $w_{i} \in \Omega_{n}$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-4.jpg?height=612&width=2243&top_left_y=20268&top_left_x=58&polygon=1864,0,1835,29,1660,29,1631,59,1515,59,1485,29,466,29,437,59,204,59,175,29,88,29,58,59,29,59,29,146,0,175,0,524,29,554,29,583,58,612,321,612,350,583,437,583,554,466,845,466,874,437,1019,437,1049,408,1078,408,1107,437,1136,437,1165,408,1311,408,1340,437,1951,437,1980,408,2068,408,2097,437,2213,437,2213,379,2243,350,2243,321,2213,291,2213,59,2184,59,2126,0,2097,0,2068,29,2039,29,2010,0) for $A_{n} \in \mathcal{F}_{n}$ then $R=\square \mathcal{F}_{n} A_{n}$ Dearte $S$ to be the fudg genertad by $P$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-4.jpg?height=437&width=2243&top_left_y=21433&top_left_x=58&polygon=1515,0,1485,29,699,29,670,58,29,58,0,87,0,350,58,408,146,408,175,437,204,437,233,408,321,408,379,350,554,350,583,320,670,320,699,350,757,350,787,320,903,320,932,350,990,350,1019,320,1136,320,1165,350,1398,350,1427,379,1485,379,1515,350,1573,350,1602,320,1718,320,1748,291,1777,291,1806,262,1893,262,1922,291,2039,291,2068,320,2184,320,2213,291,2213,262,2243,233,2243,87,2213,58,2213,29,2184,0)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-4.jpg?height=379&width=1748&top_left_y=21695&top_left_x=58) probations $\tilde{\omega}_{n}: \Omega \rightarrow \Omega_{n}$ war $\tilde{\omega}_{n}\left(\left(\omega_{1}, \ldots\right)\right)=\omega_{n}$ theN "
Notaton: ( and $A \in \lambda \times y, x \in \mathbb{X}$ an $\underline{A}_{x}=\{y \in y:(x, y) \in A\}$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-4.jpg?height=13163&width=2301&top_left_y=24083&top_left_x=0)
$\|f\|_{\infty}$
$=\inf \{f \in[-\infty, \infty]: \mu(\{|f|>+\})=0\}$
- ess spe \{lfl\}
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-5.jpg?height=598&width=664&top_left_y=2540&top_left_x=1436)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-5.jpg?height=222&width=796&top_left_y=2872&top_left_x=0)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-5.jpg?height=288&width=1392&top_left_y=3225&top_left_x=44&polygon=1348,0,1392,45,1392,243,663,288,331,288,22,266,0,243,0,45,795,0)
, 1 and $\infty$ an anyouges.
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-5.jpg?height=287&width=1966&top_left_y=3712&top_left_x=0&polygon=1303,0,1679,22,1944,44,1966,66,1966,243,1922,265,1767,287,839,287,44,265,0,176,0,66,44,22,220,0)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-5.jpg?height=244&width=1105&top_left_y=3910&top_left_x=0)
ten $\{f\}<\}=\{\omega \in \Omega: f(\omega)>i\}$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-5.jpg?height=266&width=1481&top_left_y=4352&top_left_x=265)
$\mu(2 f f+3) \leqslant \frac{1}{4} \int f d \mu$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-5.jpg?height=266&width=1238&top_left_y=4905&top_left_x=44)
$t_{p}(2 \leqslant z+3)=+\int \eta_{p+x} d x$ $\_\_\_\_$
Ant
$\leq \int f d \mu$
$\Rightarrow \mu(\varepsilon f+t) \leq \frac{1}{+} \int f d \mu$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-5.jpg?height=266&width=1503&top_left_y=5965&top_left_x=44)
(1)7) clexysten's They
for of measurable, and $m \in \mathbb{R}$
$\mu(\{|f-m|+t\}) \leq t^{-2} \int(f-m)^{2} d \mu$
(2) chenate's In
for $\quad f \quad$ mesuruble, $\eta \in \mathbb{R}$
$\quad \mu(\varepsilon f+3) \leq e^{-\eta+} \int e^{\eta f} d \mu$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-5.jpg?height=399&width=1216&top_left_y=7821&top_left_x=44)
Let $I \subseteq \mathbb{R}$ be an internal ; a fultion $\phi: I \rightarrow \mathbb{R}$ is convex
$\forall t \in[0,1]$ and all $x, y=\sum \phi(t x+(1-t) y) \leq \phi(x)+(1-t) \phi(y)$
$\_\_\_\_$ "Seciart line "
amogs on top"
Therem (Jensen's Treq)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-5.jpg?height=421&width=2232&top_left_y=9191&top_left_x=0)
have $\quad \phi\left(\int x d r\right) \leq \int \phi(x) d r$
T.e. $\phi(\mathbb{E} x) \leq \mathbb{E}[\phi(x)]$

Proof : For Some $c<I$, if $x=c$ p.a.c. Hen we're don
edese, let $M=E X$ is in the Intentor of $I$
Thenime an choose $a s \in \mathbb{R}$ s.t. $\phi(x) \geqslant a x+b \quad \forall x \in I$
and $\phi(m)=a m+b$
Then $\phi(x) \geq a x+b(x)$
Then $\phi(x) \geq a x+b[x]$
$\therefore \phi(\mathbb{E} x)=a m+b=\mathbb{E}[a n+b] \leq \mathbb{E}[\phi(x)]$
Lostly, is $\mathbb{C}[\phi(x)]$ wh defined? ${ }^{\text {® }}$ e $\neq \infty$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-5.jpg?height=288&width=885&top_left_y=11622&top_left_x=287)
and $\phi^{-}(x) \leq|a||x|+|b|$
tenne, $\mathbb{E}[\phi(x)] \leq|a| \mathbb{E}|x|+|b|<\infty$
Holder + Mamboush
Trevem: $($ Helder $)$
Holder + Minhoush
Then $\|f g\|_{2} \leq\|f\|_{p}\|g\|_{q}$
Proof: If etter int
Proof: If cuther $\| f l p=0, \infty$ or $\|q\|_{1}=0, \infty$ the
Hence, let $0<\|f\|_{p}<\infty$ then we can "romolise" whog $\| f l l_{p}=1$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-5.jpg?height=376&width=1393&top_left_y=13301&top_left_x=110)
$\square \mathbb{P}(A)=\int_{A} f d \mu$
Then, wia Tersen and moting thit $2(p-1)=p$
$\|f g\|_{1}=\int(f g) d \mu=\int \frac{1 g \mid}{f\left(r^{-1}\right.} \frac{1}{f(f \mid>0}|f|^{p} d \mu$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-5.jpg?height=354&width=1636&top_left_y=14185&top_left_x=44)
$\left[\leqslant\left[\int x^{2} x\right]^{2}-1+4 \ln 4 y^{2} l\right.$

Note: $p=q=2$, m have canchy Schiner $z$ Ineq
Therem: (Minhershish's Ing)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-5.jpg?height=398&width=2299&top_left_y=16505&top_left_x=44&polygon=1503,0,1481,22,1127,22,1105,44,1083,44,1061,66,818,66,796,89,796,111,774,133,729,133,707,155,641,155,619,177,597,177,575,199,177,199,155,221,22,221,0,243,0,376,22,398,155,398,177,376,221,376,243,354,486,354,508,332,508,243,530,221,1171,221,1193,199,1282,199,1304,177,1326,199,1348,177,2276,177,2298,155,2298,66,2276,44,2276,22,1967,22,1945,0)
Root: If $\|f\|_{p}=\infty$ or $\| g l_{p}=\infty$ the done
If $\|f+g\|_{p}=0$ done
Mffillp =0 dore
otherwise, $|f+g|^{p}=2^{p}\left|\frac{\left.\right|^{f+g}}{2}\right|$
$\leqslant 2^{p}\left(\frac{1}{2}|f|^{p}+\frac{1}{2}|g| p\right)$
$2^{p-1}\left(1 f_{i}^{p}+g_{g^{p}} \rho^{p}\right)$
$\int|f+g|{ }^{p} d \mu \leq z^{p-1} \int|f|^{p} d \mu+\left.z^{p-1} \int \lg \right|^{p} d \mu<\infty$
$\therefore f+g \in L(\Omega, F, \mu)$
Assariy, $\|f+g\|_{p}>0$ and $p>1$ and $p, z$ congyotes then
$\left\|\left.f_{f+g}\right|^{r^{-1}}\right\|_{q}=\left[\int\left(f_{f}\right]^{(f-)} q_{q}\right]^{\frac{1}{2}}$
$\left[\int|i+g|^{p} d \mu\right] \frac{t-1}{r}$
$\left\|f^{+} J\right\|_{p}^{+-1}$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-5.jpg?height=244&width=2122&top_left_y=19311&top_left_x=132)
Ulfeglis?
$+\|q\|_{+}\left\|p_{t y} p^{p^{-1}}\right\|_{2}$
$\|f\|_{p}\|\mathrm{Arg}\|_{p}^{+}+\|\mathrm{g}\|_{p}^{-}\|\mathrm{Alg}\|_{1}^{+}$
$\|f+g\|_{f}^{[r-1]}$
Theorem (Apper in $c^{p}$ spout)
Let $(\Omega, \mathcal{F}, \mu)$ an a reaste space, $A \Omega$ :
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-5.jpg?height=309&width=2320&top_left_y=20659&top_left_x=44&polygon=2320,0,2320,265,1215,309,199,309,88,287,0,265,0,154,22,110,88,66,176,44,618,22,2298,0) \begin{tabular}{l}
$-d A_{i} \in A_{\text {s.t. }} A_{i} \lambda \Omega$ \\
\hline
\end{tabular}
For $p \in[1, \infty), V_{0} \subset L^{p}$ and $\forall f \in L^{p}, \forall z>0 \quad \exists \in V_{0}$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-5.jpg?height=377&width=2321&top_left_y=21520&top_left_x=44)
$\mu(A)^{\frac{1}{c}}<\Delta$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-5.jpg?height=421&width=2321&top_left_y=21984&top_left_x=44&polygon=2298,1,2298,23,2254,67,2232,67,2188,111,2166,111,2143,89,2121,111,1945,111,1922,89,1856,89,1834,111,1039,111,1017,133,972,133,950,111,442,111,420,133,89,133,66,156,44,156,22,178,44,200,44,222,22,244,22,288,0,310,0,332,22,354,22,376,44,399,111,399,133,421,177,421,199,399,265,399,287,376,354,376,376,354,663,354,685,376,751,376,774,399,796,399,818,376,862,376,884,399,1503,399,1525,376,2077,376,2099,354,2121,376,2298,376,2320,354,2320,67,2298,45)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-5.jpg?height=465&width=2216&top_left_y=22565&top_left_x=31)
$\left\|(f+g)-\left(v_{f}+v_{y}\right)\right\|_{p} \leq\left\|f-v_{f}\right\|_{p}+\left\|_{q}-v_{y}\right\|_{p}<2 \varepsilon$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-5.jpg?height=575&width=1769&top_left_y=23465&top_left_x=88)
We have $A \subset \mathcal{L}$ ad ∴ $\Omega \in \mathcal{Z}{ }_{\text {for }}$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-5.jpg?height=266&width=1305&top_left_y=24260&top_left_x=220)
Let $A=\bigcup_{j=1}^{\infty} A_{i}, G_{j}=\bigcup_{j=1}^{J} A_{i}$
Then $\varepsilon_{j} \uparrow A$ and $\left\|\frac{1}{A}-n_{B_{j}}\right\|_{i}=\Gamma\left(A \backslash B_{j}\right)^{\frac{1}{p}} \longrightarrow 0$
$\Rightarrow \mathcal{L}$ is $-\lambda$-system
⇒ Pymin $\pi-\lambda$ says $\mathcal{F}=\sigma(A) \leq z$
Now for am ron-regative $f \in L^{+}$, we construt suple
fn $=$min $\left[n, i^{-n}\right]$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-5.jpg?height=355&width=2321&top_left_y=26138&top_left_x=44&polygon=1326,1,1304,23,818,23,796,45,442,45,420,67,398,67,376,45,199,45,155,89,89,89,66,111,44,111,44,133,0,178,0,222,22,244,22,266,44,288,44,310,89,354,155,354,177,332,221,332,243,310,641,310,663,288,729,288,751,310,774,288,818,288,840,266,1017,266,1039,244,1061,266,1171,266,1193,244,1260,244,1282,266,1304,266,1326,244,1525,244,1547,222,1591,222,1613,244,1834,244,1856,266,2210,266,2232,244,2298,244,2320,222,2320,45,2298,23,1812,23,1790,1,1679,1,1657,23,1635,23,1613,1)
and $\mid f-$ m $\left._{n}\right|^{p} \leq|f|^{p} \leq$ bound
- By Domint-al convegerres $\left(f-f_{n} \| p-\left[\int\left|f-f_{n}\right|^{p} d \mu\right]^{\bar{P}} \rightarrow 0\right.$

Lasty, for guered $\Omega \notin A, L_{4}$ assumption $A_{1} \uparrow \Omega$
and $|f-f|_{A} \mid p \rightarrow 0$ pointairse and $\left.|f-f|_{A i}\right|^{p}|f|^{p} \therefore\left\|f-f p_{A}\right\|_{P}$ and $\left.|f-f|_{A i}\right|^{P} \leq|f|^{P}$

\section*{Lecture 10: Convergence of Measure}

Weale convergence of measure

\begin{tabular}{|l|l|}
\hline \multicolumn{2}{|l|}{$(\Omega, \neq)$ be a messurble space $\left\langle A_{i} S_{i=1}^{\infty}\right.$ a saq of prob medures s.t. $\mathbb{P}_{1}: \mathcal{F} \rightarrow \mathbb{R}^{+}$} \\
\hline \multicolumn{2}{|l|}{$Q:$ What does $\mathbb{P}_{i} \rightarrow \mathbb{P}$ mean?} \\
\hline \multicolumn{2}{|l|}{\begin{tabular}{l}
e.g. Maybe we want $\mathbb{P}_{i}(A) \rightarrow \mathbb{P}(A) \forall A \in \neq$ ? \\
$\uparrow$ \\
,
\end{tabular}} \\
\hline \multicolumn{2}{|l|}{\multirow[b]{2}{*}{Let $s$ be $a$ metric space $\rho$ be the Bord $\sigma$-fierd on $s$}} \\
\hline & \\
\hline \multicolumn{2}{|c|}{\begin{tabular}{l}
Then, we write $\mathbb{P}_{i} \Rightarrow \mathbb{P}$ for wead annegerves if $\int f d \mathbb{P}_{i} \rightarrow \int f d \mathbb{P}$ \\
$\forall f \in C_{B}^{r}(\mathbb{R})$, all continuous bounded funtions $S \rightarrow \mathbb{R}$
\end{tabular}} \\
\hline \multicolumn{2}{|c|}{$d: s \times s \rightarrow R^{+}$} \\
\hline if [ $\mathbb{P}_{3}$ ] converges then $\mathbb{R}$ prosusely, for a finite collection & \begin{tabular}{l}
$\_\_\_\_$ \\
thin
\end{tabular} \\
\hline an $\varepsilon$-reighborhand, any $\varepsilon>0$, of $P$ to $e$ all $\mathbb{Q}$ Q & \\
\hline $$\left|\int f_{i} d t-f_{f i} d Q\right|<\varepsilon \quad \theta_{i}=1, \ldots$$ & $$\binom{Q_{0}}{g^{R}}$$
$$\cdot^{\mathbb{R}}$$ \\
\hline
\end{tabular}

\begin{tabular}{|l|}
\hline Theorem: (Portmantear) \\
\hline Proof: (1) ⇒(2). If convergence holds $\forall f \in C_{\beta}(\mathbb{R})$ then it holds for $f \in C_{B}(\mathbb{R})$ that are also uniformy con 17 $(2) \rightarrow(3)$. \\
\hline \\
\hline \begin{tabular}{l}
∴ tabe limsup and $\varepsilon \downarrow 0$ \\
$\limsup \mathbb{R}_{i}(c) \leq \mathbb{P}(c)$
\end{tabular} \\
\hline \begin{tabular}{l}
(3) ⇒ (1) Let $f \in C_{B}(\mathbb{R})$. hoal is to show limsup $\int f d \mathbb{P} i \leq \int f d \mathbb{P}$ and similarly for limint: \\
$f$ is bounded by assumption $\therefore$ we can shift and scile it whoc assure $0<f<1$
\end{tabular} \\
\hline \begin{tabular}{l}
feemarl: - wed cown $\mathbb{P}_{i} \Rightarrow \mathbb{P}, f \in c_{B}(\mathbb{R})$ \\
- Radon puetric : sup $\left[\int f d \mathbb{P}_{i}-\int f d \mathbb{P}\right] \rightarrow 0$ all continues
$$f: S \rightarrow[-1,1]$$ \\
- if we take all $f$ measurble $f: 5 \rightarrow[-1,1]$ " Total variefion" \\
- if $f$ are Lopschitz, constact $I$ "how quilly does the converginece conv in 1 -wasserstein \\
occor \\
→ vieful in optinet trevispont theory \& ML
\end{tabular} \\
\hline \\
\hline \begin{tabular}{l}
Cow of Random Vaviobles \\
Let \\
$(\Omega, \mathcal{Z}, \mu)$ be a Probability space $c s, \rho)$ be a retric space Then a rarelom variable $x: \Omega \rightarrow S$ detires a measure on $\rho$
$$\mathbb{P}(A)==\mu\left(x^{-1}(A)\right), A \in \rho$$
\end{tabular} \\
\hline \\
\hline \begin{tabular}{l}
For a $\operatorname{seg}\left\{x_{i}\right\}_{i=1}^{\infty}, x_{i} \xrightarrow{d} x$ (in distribution) if $\mathbb{P}_{i} \rightarrow \mathbb{P}$ \\
For Notetion: $\mathbb{E}[x]=\int_{\Omega} x(w) d \frac{1}{\mu}(w)=\int_{5} x d \mathbb{P}(x)$ \\
also $\mathbb{P}_{i}(A):=\mathbb{P}_{i}\left(x_{i} \in A\right)$ \\
Note: \\
The pootmentein theorem \\
can he rewritten for Roudom Variables \\
Convergence in Probability \\
$X=\xrightarrow{\mathbb{P}} X$, if for all $\varepsilon>0$,
\end{tabular} \\
\hline \\
\hline
\end{tabular}

\begin{tabular}{|l|}
\hline $$\mu\left(\left\{\omega \in \Omega: d\left(X_{i}(\omega) ; X(\omega) ; \tau\right\}\right) \rightarrow 0\right.$$ \\
\hline In short, $\mathbb{P}\left(d\left(x_{1}, x\right)>\varepsilon\right) \rightarrow 0$. Convergence in posberility is closefy recated to $x$ Convergencie Almost Survely \\
\hline $$x_{i} \xrightarrow{a, s} x \quad \text { if } \mu\left(\left[\omega \in \Omega: x_{i}(\omega) \rightarrow x(\omega)\right]\right)=1$$ \\
\hline i.e. pointaite onergence almost everyobtere/ surely andayst. robabilist \\
\hline $$\mu\left(\left\{\omega \in \Omega: x_{i}(\omega) \nrightarrow x(\omega)\right\}\right)=0$$ \\
\hline Noter: Hu rettir $d$ does not apper here Convergence $L L^{P}$ \\
\hline $$x_{i}^{L P} x, \mathbb{E}\left[d\left(x_{i} ; x\right)^{p}\right]=\int_{\Omega} d \underbrace{\left(x_{i}(\omega), x(\omega)\right)}_{\text {finfor }})_{\Omega \rightarrow R^{p}}^{p} \cdot d_{\mu}(\omega) \rightarrow 0$$ \\
\hline $$\text { if } \rho=\mathbb{R} \text { then } \int\left|x_{i}-x\right|^{p} d \mu \rightarrow 0$$ \\
\hline Hieviarchy of Convegerve \\
\hline
\end{tabular}

Conv a.s ⇒ conv prob
conv prob → conv dist

any $p \in[1, \infty]$, conv $L^{p} \Rightarrow$ conv prob
But convegarie $9.9 . \longleftrightarrow$ conv $L^{p}$
Borel - Contelli Lemmas
$\frac{\text { Borel-Contelli Lemmas }}{\text { Let }(\Omega, F, \mu) L e \text { a probability space }\left\{A_{i}\right\}_{i=1}^{\infty}, A_{i} \in Z}$
them
$\therefore \liminf A_{i}=\bigcup_{j=1}^{\infty} a_{j>i} A_{j}$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-6.jpg?height=263&width=2289&top_left_y=17849&top_left_x=34) → we also say $A_{i}$ everaltuly", $A_{i}$ ev. for limitot $A_{i}$
because $\exists w \in \mathbb{N}$ s.t. $\forall n \geqslant N \quad w \in A_{n}$
Theorem (1 $1^{\text {st }}$ Borel-Contelii)
Let $\left\{A_{i}\right\}_{j=1}^{\infty}$ with $A_{i} \in F$ if $\Sigma_{i=1}^{\infty} \mu\left(A_{i}\right)<\infty$ then $\mu\left(\right.$ mimspep $\left.A_{i}\right)=0$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-6.jpg?height=228&width=2271&top_left_y=18652&top_left_x=0) Proof: As
as $\begin{aligned} \therefore \mu \rightarrow \infty\left(\limsup A_{i}\right) & =\mu\left(\bigcap_{i=1}^{\infty} \cup A_{i}\right) \\ & \leq \mu\left(\bigcup_{j>i} A_{i}\right) \\ & \leq \sum_{j>i} \mu\left(A_{i}\right) \rightarrow 0 \text { as → } \square\end{aligned}$

\begin{tabular}{|l|l|}
\hline Thorem $\left(2^{\text {rt }}\right.$ Bord-Cantellit Iomm $)$ & \\
\hline Let $\left\{A_{i}\right\}_{i=1}^{\infty}$ be redepreduct and $A_{i} \in \mathcal{F}$. Then $A_{i=1}^{\infty} \sum_{i=1}^{\infty}\left(A_{i}\right)=\infty$ & \\
\hline then $\mu\left(\right.$ cims $\left.p_{1}, p_{1}\right)=1$ & \\
\hline Prod: Note thet $1-t \leq e^{-t}, \forall t \in \mathbb{R}$ & Ove con chech that if $\left\{A_{1}\right\}_{i}^{\infty}=1$ are \\
\hline independent then $\left[A_{i}^{c}\right]_{i}^{\infty}$ ore independent & \\
\hline - for any $i \in \mathbb{N}$ and $k \geqslant 1$ & $$\begin{aligned}
\mu\left(\prod_{j=1}^{k} A_{j}^{c}\right)^{\prime} & =\pi\left[1-\mu\left(A_{j}^{i}\right)\right] \\
& \leq \exp \left[-\sum_{j}^{k} \mu\left(A_{j}\right)\right]
\end{aligned}$$ \\
\hline & \\
\hline Take $K \rightarrow \infty$ and $R H S \rightarrow 0$ & \\
\hline $$\begin{aligned}
\therefore \mu\left(\bigcap_{j A^{c}}^{c}\right) & =0 \quad \forall i \\
\therefore \mu\left(\limsup A_{i}\right) & =\mu\left(\bigcup_{i=1}^{\infty} \nu_{j i} A_{j}\right) \\
& =1_{-\mu\left(\bigcup_{i=1}^{\infty} A_{j}^{c} A_{j}^{c}\right)}=1
\end{aligned}$$ & \\
\hline
\end{tabular}

Lecture II: The story Law of Lage Number $01 / 05 / 26$ Let $\left\{x_{i}\right\}_{j=1}^{\infty}$ be restern variables from $(\Omega, F, \mathbb{P})$ to $(\mathbb{R}, B)$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-7.jpg?height=169&width=812&top_left_y=338&top_left_x=33)
- $\mathbb{P}(x \in A)=\mathbb{P}\left(\sum \omega G \Omega: x(\omega) \in A()\right.$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-7.jpg?height=440&width=1454&top_left_y=743&top_left_x=287)
$E X=\int x(0) d \mathbb{R}$
$S_{n}=\sum_{n} x_{n}$
Defrition (todesendence)
$x$ and $y$ on the same polabolitity space $(\Omega, F, \mathbb{P})$ wit with possibly different $(x, x)$ and $(y, y)$.

\begin{tabular}{|l|}
\hline Then we say that $x$ all $y$ are indepindent if $\mathbb{P}(\{x \in A\} \cap\{y \in B))=\mathbb{P}(x \in A) \mathbb{P}(y \in B) \quad \forall A \in \mathbb{X}$ and $B \in \mathcal{Y}$ \\
\hline This con be extardel to finite overations $\left[x_{i}\right\}_{i=1}^{\wedge}$ with \\
\hline - Also, $\left\{x_{i}\right\}_{i=1}^{a}$ s independent if every finite set of $X_{i}$ 's is independent → similor to what me saw for j-freble \\
\hline Then $x$ and $\varphi$ ore solependert if and only if $\sigma$-firely $\sigma(x)$ and $\sigma(y)$ are molepredult. \\
\hline Identically Distributed "you want the RV's to have the same distribution \\
\hline \\
\hline Theorem (Weal las of lage numbers) \\
\hline het $(\Omega, F ; 1, P)$ be a probability space, $\left\{x_{i}\right\}_{i=1}^{\infty}$ se renotom vateles from $\Omega \rightarrow \mathbb{R}$ s.t. $\mathbb{E} X_{i}=c \in \mathbb{R}$ and $\mathbb{E} X^{2}=1 \quad \forall i=1, a$ \\
\hline and $\mathbb{4}\left[\left(x_{i}-c\right)\left(x_{j}-c\right)\right]=0 \quad \forall i \neq j$ Then $\frac{S_{n}}{n} \xrightarrow{\mathbb{R}} c$ in porbability![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-7.jpg?height=427\&width=657\&top_left_y=5415\&top_left_x=1618) \\
\hline Proot: $w \log c=0$, therwise replace $x_{i}$ with $x_{i}=c$ Then for any $t>0$, Chelysher says, \\
\hline \\
\hline $$\mathbb{P}\left(\frac{\left|S_{n}\right|}{n} \geqslant t\right) \leq \frac{\mathbb{E} S_{n}^{2}}{t^{2} n^{2}}=\sum_{1, j=1}^{n} \frac{\mathbb{E}\left[x_{i} x_{j}\right]}{t^{2} n^{2}}=\frac{1}{n t^{2}} \rightarrow 0$$ \\
\hline In the rext theorm, me require independent $X_{1}$ 's' me also need the voriance
$$V_{o r}(x)=\int(x-\mathbb{E} x)^{2} d \mathbb{P}(\omega)$$ \\
\hline Thomem (Strong Loge Nonbers) \\
\hline het $\left\{x_{i}\right\}_{i}^{\infty}=$ rentom veriables from $\Omega \rightarrow \mathbb{R}$, \\
\hline \begin{tabular}{l}
If $\mathbb{E}\left|x_{i}\right|<\infty$, then $\frac{S_{n}}{n} \xrightarrow{\text { a.s. }} c$ for $c=\mathbb{E} x_{1}$ (a) - Also, if $E\left|X_{i}\right|=\infty$, then $\frac{S_{n}}{n}$ does not converge to![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-7.jpg?height=299\&width=351\&top_left_y=8004\&top_left_x=1992) \\
finite limit. ${ }^{(2)}$
\end{tabular} \\
\hline \\
\hline Proof: (2) Assume $n^{-1} S_{n} \rightarrow C \in \mathbb{R}$ att afo $\mathbb{E}\left|X_{1}\right|=\infty$ Note that $\frac{x_{n}}{n}=\frac{S_{n}-S_{n-1}}{n} \rightarrow 0$
$$\frac{S_{n}}{n}-\frac{S_{n-1}}{n} \rightarrow c-c=0$$ \\
\hline Since $\mathbb{E}\left|x_{i}\right|=\infty$ then $\sum_{n=0}^{\infty} \mathbb{P}\left(x_{i}>n\right)=\infty$, twe Borel - cartell: \\
\hline says that $\left|x_{n}\right|>n$ for 10 . Thus: $\mathbb{P}\left(\left\{\omega \in \Omega: \frac{S_{n}-S_{n-1}}{n} \rightarrow 0\right\}\right)=0$ manituely often
$$\therefore n^{-1} S_{n}+c \Rightarrow \mathbb{R}$$![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-7.jpg?height=325\&width=487\&top_left_y=9699\&top_left_x=1822) \\
\hline (1) Assume $\mathbb{E} x_{i}=c \in \mathbb{R}$ - wilog $x_{1} \geqslant 0 \quad \theta_{1}$ else write $x=x^{+}-x^{-}$ and do everything for $x^{+}, x^{-}$ \\
\hline \\
\hline Also, independence of $x, y$ implies Independence of $x^{t}$ and $\varphi^{t}$ Also, $F$ devotes the low of $X$ i.e $F(x)=\mathbb{P}(X \leq x)$ \\
\hline \\
\hline \\
\hline \\
\hline $$\begin{aligned}
& \sum_{n=1}^{\infty} k_{n}^{-2} \mathbb{1}_{k n \geq a} \leq 4 \sum_{n=1}^{\infty} \delta^{-2 n} \mathbb{1}_{\delta^{n} \geq i} \leq \frac{4}{2^{2}\left(1-f^{-2}\right)} \leq c_{0} i^{-2} c^{-2} \\
& \text { for Some } c_{0}>0
\end{aligned}$$ \\
\hline
\end{tabular}
$\sum_{i=k+1}^{\infty} i^{2}<\int_{k}^{-1} x^{-2} d x=\frac{1}{k}$
Now, for any $t>0, \exists c>0$ depereling ony on $t$ and $\delta$ s.t.
$\sum_{n=1}^{\infty} \mathbb{P}\left(\left|T_{k_{n}}-\mathbb{E} T_{k_{n}}\right|>+k_{n}\right) \leq c_{1} \sum_{n=1}^{\infty} k_{n}^{-2} \operatorname{Var}\left(T_{k_{n}}\right)$
$=c_{1} \sum_{n=1}^{\infty} \frac{1}{k_{n}^{2}} \sum_{i=1}^{k_{n}} V_{r}\left(Y_{i}\right)$
$=c_{i}^{\infty} \sum_{i=1}^{\infty}\left[\begin{array}{ll}V_{i}-\left(y_{i}\right) & \sum_{k_{n} ?} \\ k_{n}^{2}\end{array}\right]$
$\leq c_{2} \sum_{1}^{\infty} c^{-2} \mathbb{E} Y_{1}^{2}$
$=c_{i} \sum_{i=1}^{\infty}-2 \int_{0}^{\infty} x^{2} d F(x)$
$\left.=c_{2} \sum_{i=1}^{n} \sum_{n=0}^{i-1} \int_{k}^{k+1} x^{2} d F(x)\right\}$
$\leq c_{3} \sum_{k=0}^{\infty} \frac{1}{k+1} \int_{k}^{k+1} x^{2} d F(x)$
$\leq c_{3} \sum_{k=0}^{n} \int_{k}^{k+1} x d F(x)$
$\leq c_{3} \sum_{k=0}^{\infty} e^{-} x_{i}<\infty$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-7.jpg?height=423&width=863&top_left_y=15379&top_left_x=1487)

In the we, $\sum_{n=1}^{\infty} \mathbb{R}\left(\left|T_{m}-\varepsilon_{T_{m}}\right| \geqslant \mathrm{Fth}_{n}\right)<\infty$
Borel-Cantelli (RC) says $\frac{1}{K_{n}}\left|T_{u_{n}}-\mathbb{E} T_{\text {in }}\right| \xrightarrow{a . s_{n}} 0$ Since $\mathbb{E} Y_{n} \uparrow \mathbb{E} X_{i}$ we have that $\quad$ Ubrip in convergent in $K_{n}^{-1} \mathbb{E} T_{k_{n}} \uparrow \mathbb{E} X_{i}$ as $n \rightarrow \infty \quad$ probability to almost currely $\therefore K_{n}^{-1} T_{k n} \xrightarrow{0 . s} \mathbb{E} X_{i}$
To get beel to $S_{n}$, note that for $x, \mathbb{E} X<\infty \Longrightarrow \sum_{i=0}^{\infty} \mathbb{T}$

Then $\sum_{i=1}^{\infty} \mathbb{P}\left(x_{i} \neq y_{i}\right)=\sum_{i=1}^{\infty} \mathbb{P}\left(x_{i}>i\right)<\infty$
BC says $\pi\left(\operatorname{limap}\left\{x_{i} \neq y_{i}\right\}\right)=0$
Hence, for $z$ longe enough $x_{i}=y_{i}$ a.s.
het large enough rean all $i>M(\omega) \left(\right.$ i.e. $\left.x_{i}(\omega)=y_{i}(\omega) \theta_{i}>m(\omega)\right)$
Forthermore, $K_{n}^{-1} S_{m(\omega)} \rightarrow 0$ and $K_{n}^{-1} T_{m(\omega)} \rightarrow 0$
as $n \rightarrow \infty$ terms's where
negigible
$$\therefore K_{n}^{-1} S_{K_{n}} \xrightarrow{\text { a.s. }} Z X_{i} \quad \therefore \text { we have almost sire (a.s.) aniegene }$$

Finally, $\frac{k_{n}+1}{k} \rightarrow \delta$ as $n \rightarrow \infty$
$$k_{n} \quad \text { An lorge enough st. } 1 \leq \frac{k_{n+1}}{k_{n}}<\delta^{2}$$

This, for $k_{n}<i<k_{n+1}$
$$\frac{S_{k_{n}}}{k_{n}} \leq \delta^{2} \frac{S_{1}}{i} \leq \delta^{n} \frac{S_{k n+1}}{k_{n+1}}$$
$\delta^{-2} \mathbb{E} x_{1} \leq \liminf _{i \rightarrow \infty} \frac{S_{i}}{i} \leq \limsup _{i \rightarrow \infty} \frac{S_{i}}{i} \leq \delta^{2} \mathbb{E} x_{i}$
$\frac{\text { Lect }}{\text { orem }}$
12: Centrel Limit Theorem 04/05/26

\begin{tabular}{|l|}
\hline \multirow{2}{*}{\begin{tabular}{l}
Prohorov's Theorem "Heren bout congaretress" \\
Defn : (Onform Tghtress)
\end{tabular}} \\
\hline \multirow[b]{2}{*}{![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-8.jpg?height=350\&width=2241\&top_left_y=471\&top_left_x=45)} \\
\hline \\
\hline uniformy tight if $\forall \varepsilon>0$ JK compart s.t. $\mu_{i}\left(K_{\varepsilon}\right)>1-\varepsilon$ \\
\hline Theorem (Proborous) $\leftarrow$ couple different versions \\
\hline \\
\hline $\Rightarrow \mu$ to some $\mu$ depe \\
\hline \\
\hline Apposition: \\
\hline If $\left\{\mu_{i}\right\}_{j=1}^{\infty}$ and $\left(\mu\right.$ ore prob reasures st. $\theta_{\text {ik }} \mu_{i_{k}}=\mu$ \\
\hline \multirow{2}{*}{\begin{tabular}{l}
Proof: \\
Assume not, then $\exists f t_{c}$ b ret. $\int f d \mu \rightarrow \int f d \mu \exists \varepsilon>0 \quad \exists_{k}$ s.t. $\left(\int f d_{\mu_{k}}-\int f d_{\mu} / 2 \varepsilon \forall \varepsilon\right.$
\end{tabular}} \\
\hline \\
\hline However, $\exists i_{k_{r}}$ s.t. $\mu_{k_{r}} \Rightarrow \mu$ confradietion \\
\hline Defn (Gonssi- measire) \\
\hline A Broel measure $\gamma$ on $(\mathbb{R}, T B)$ id said to be Cearston with
mean $m$ and voriance $g^{2}$ if \\
\hline $\gamma((a, b])=\frac{1}{\sigma \sqrt{2 \pi}} \int_{a}^{b} \exp \left[-\frac{1}{2 \sigma^{2}}(x-m)^{2}\right] d \lambda(x)$ \\
\hline \multirow{2}{*}{\begin{tabular}{l}
Also, for $\sigma=0, \gamma=\delta_{m}$ (Dirac measure) we say $\gamma$ is a
deoperente Coussian mesove. \\
degrented Coussion resolve.
\end{tabular}} \\
\hline \\
\hline Defn (in $\mathbb{R}^{a}$ ) \\
\hline $\gamma$ on $\left(\mathbb{R}^{-1}, \beta\right)$ is Caussian it Ulinem furbions $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$ the rabled nerred $\gamma \circ f^{-1}$ or $(R, \beta)$ is Gaussian. Ary thear combination of Coussion \\
\hline Defn ( $R . V$ ) \\
\hline A RV. Z from $(\Omega, F, \mu)$ to $\left(R^{l}, \beta\right)$ is Causitan if $\gamma:=\mu \cdot Z^{-1}$ is a caussian measure. \\
\hline For rectors $u, v \in \mathbb{R}^{\alpha}$, the inver product $\langle u, v\rangle=\left\{4 v_{i}\right.$ \\
\hline \multirow{2}{*}{\begin{tabular}{l}
$141^{2}=\langle 4,4\rangle$ \\
Chereforsistic foretion
\end{tabular}} \\
\hline \\
\hline $A$ probability reasure $\mu$ on $\left(\mathbb{R}{ }^{\alpha}, \beta\right)$ the characteristic forefor is $\mu \tilde{\mu}: \mathbb{R}^{x} \rightarrow \mathbb{C}$ \\
\hline $$\begin{gathered}
\tilde{\mu}(\nu):=\int \exp \{i<x, \nu>\} d \mu(x) \\
\tilde{r}(\text { to sealling } \\
=\sqrt{1}
\end{gathered}$$ \\
\hline \\
\hline \\
\hline $p(x)$ is the probability density funtion for $\gamma$ a.e. \\
\hline \begin{tabular}{l}
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-8.jpg?height=136\&width=698\&top_left_y=7900\&top_left_x=1621) \\
Convolution fintions"
\end{tabular} \\
\hline \multirow{3}{*}{![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-8.jpg?height=598\&width=2308\&top_left_y=8148\&top_left_x=56)} \\
\hline \\
\hline \\
\hline ![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-8.jpg?height=146\&width=2265\&top_left_y=8809\&top_left_x=44) \\
\hline ![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-8.jpg?height=349\&width=935\&top_left_y=9004\&top_left_x=112) \\
\hline Also, for indelpendent $C V^{\prime} \backslash X, Y$ with megures $\mu_{\nu} \nu$ X $X Y$ has reduced measure $\mu * \nu$ \\
\hline Theoren (uniqueness of $\sim$ ) \\
\hline Proof : Let $\gamma_{j}$ be a mean zero Gairsia measure a $\mathbb{R}^{d}$ with co-vertance $\sigma^{2} \mathbb{P}_{d}$. \\
\hline Devote $\mu^{(\sigma)}=\mu * \gamma_{0}$ and Same for $\nu^{(\sigma)}$. \\
\hline It cal be shown that the correspondiry sensity functions for $\mu^{(\sigma)}$ and $\nu^{(\sigma)}$ are \\
\hline \multirow{5}{*}{\begin{tabular}{l}
$q^{(\sigma)}(u)=\frac{1}{(2 \pi)^{d}} \int \tilde{\nu}(t)$ exp $\left[-i<x, t>-\frac{1}{2} \sigma^{2}|t|^{2}\right] d l(t)$
$$\begin{aligned}
& \text { Then } \mu^{(\sigma)} \text { comer from } x+\sigma z \\
& \therefore x+\sigma z \xrightarrow{a .5} x \text { as ovo}
\end{aligned}$$ \\
∴ onv in probability
\end{tabular}} \\
\hline \\
\hline \\
\hline \\
\hline \\
\hline \multirow[t]{2}{*}{∴ connergence in dotribution $\mu^{(\sigma)} \Rightarrow \mu$ as $\sigma$ Lo} \\
\hline \multirow[b]{2}{*}{Theoren (CCT) (requires finite record monent)} \\
\hline \\
\hline Let $(\Omega, F, \mu)$ be a prob space $\left\{x_{n}\right\}_{n=1}^{\infty}$ are ind \\
\hline rancelom veriables a ( $\mathbb{R}^{d}, 13$ ) s.t. $\mathbb{E} x_{n}=0,\left.\sigma_{( }^{3} x_{n}\right|^{2}<\infty$ \\
\hline Let $S_{n}=\sum_{j=1}^{n} x_{j}$, then $n^{-\frac{1}{2}} S_{n} \rightarrow z$, where $z$ is Caussion \\
\hline \\
\hline \\
\hline
\end{tabular}
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-8.jpg?height=1658&width=2338&top_left_y=14547&top_left_x=0)

\begin{tabular}{|l|l|l|}
\hline \multicolumn{3}{|l|}{\begin{tabular}{l}
By the prop form above $\mu_{i} \Rightarrow \mu$ \\
口
\end{tabular}} \\
\hline \multicolumn{3}{|c|}{\multirow{3}{*}{\begin{tabular}{l}
$\_\_\_\_$ "Cherafferistic \\
$R v^{\prime} s^{\prime} x_{j} \rightarrow$ no non to \\
cheracterstic
funfion of Coursia
\end{tabular}}} \\
\hline & & \\
\hline & & \\
\hline \multicolumn{3}{|l|}{\begin{tabular}{l}
TLUS Turs \\
$\mathbb{E}\left|n^{-\frac{1}{2}} S_{n}\right|^{2}=\frac{1}{n} \mathbb{E}\left[\sum_{j, k=1}^{\{ }\left\langle x_{j}, x_{a}\right\rangle\right]$
\end{tabular}} \\
\hline \multicolumn{3}{|c|}{"Mator iretion" $=\mathbb{E}\left|x_{j}\right|^{2}$} \\
\hline \multicolumn{3}{|l|}{En any $\varepsilon>0 \quad \exists m_{\varepsilon}>0$ s.t. $\frac{E\left(x_{j}\right)^{2}}{m_{\varepsilon}^{2}}<\varepsilon$} \\
\hline ![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-8.jpg?height=210\&width=1118\&top_left_y=18001\&top_left_x=41) & & \\
\hline $$\mathbb{P}\left(\left|n^{\frac{-1}{2}} \Sigma_{n}\right|>n_{\varepsilon}\right)<\varepsilon$$ & ![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-8.jpg?height=224\&width=545\&top_left_y=18196\&top_left_x=1688) & ![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-8.jpg?height=266\&width=545\&top_left_y=18168\&top_left_x=1674) \\
\hline \multicolumn{3}{|l|}{$$\therefore \text { Seq } n^{\frac{-1}{2}} S_{n} \mapsto \quad u .7$$} \\
\hline \multicolumn{3}{|l|}{![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-8.jpg?height=168\&width=1969\&top_left_y=18782\&top_left_x=55) with $\mathbb{E}\left\langle v, x_{j}\right\rangle=0, \mathbb{E}\left\langle v, x_{j}\right\rangle^{2}<\infty$} \\
\hline \multicolumn{3}{|l|}{$\sqrt{ }$} \\
\hline $h(0)=1 \nabla L(0)=0 \nabla^{2} h(0)=-f\left[x_{j} x_{j}^{\top}\right]$
$$\text { foren: } B^{2}$$ & Geldear Nom & ![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-8.jpg?height=196\&width=615\&top_left_y=19759\&top_left_x=1423) \\
\hline \multicolumn{3}{|c|}{\multirow{3}{*}{\begin{tabular}{l}
By Taybor's Thesery, $h(v)=1$
$$-\frac{1}{2} v^{\top} 2^{v} v+o\left(v^{2}\right)$$ \\
∴ for ang fixed v, $\operatorname{Fexp}\left\{i\left\langle n^{-\frac{1}{2}} s_{n, v}\right\rangle\right\}=h\left(n^{-\frac{1}{2}} v\right)^{-}$
\end{tabular}}} \\
\hline \multicolumn{3}{|c|}{} \\
\hline & & \\
\hline
\end{tabular}

\section*{![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=308&width=1407&top_left_y=571&top_left_x=395)}
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=220&width=1605&top_left_y=747&top_left_x=351)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=243&width=1649&top_left_y=966&top_left_x=351) resienting $x$ and $a\left(T^{-1}(A)\right)$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=264&width=616&top_left_y=1538&top_left_x=351)
- set $A e^{\prime} z^{\prime \prime}$ is note: the vilution
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=265&width=572&top_left_y=1999&top_left_x=395)
$\_\_\_\_$ $\forall A \in F$
$\_\_\_\_$
$f=f \cdot T$
$=f(T \cdots)$ f vani (Cengodite)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=243&width=770&top_left_y=2724&top_left_x=373) Garres es $(0,1], B, \lambda)$ Subt mas : $T(x)=x+a$ mod $\begin{cases}x+a & x \\ x+a-1, & x+a>1\end{cases}$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=265&width=1539&top_left_y=3581&top_left_x=417) $\frac{\text { Fon Farts }}{\text { (i) } 16}$ f $\_\_\_\_$ resolve - proseming
$\int f d \mu=\int f . \bar{x} d \mu$ 2) If T as erg $\_\_\_\_$
$\_\_\_\_$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=309&width=418&top_left_y=4350&top_left_x=1516) Engotic Thorems $\Omega, F$,
$\_\_\_\_$
$\_\_\_\_$ not me obefiee $S_{n}=S_{n}(f)=f+f$

Changent
$\_\_\_\_$
$\_\_\_\_$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=397&width=1100&top_left_y=6437&top_left_x=21)
$\_\_\_\_$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=242&width=1824&top_left_y=6899&top_left_x=43&polygon=1714,0,1824,110,1824,132,1802,154,1736,176,1495,198,945,242,176,242,22,198,0,176,0,88,22,44,66,0) Samemo a
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=287&width=1231&top_left_y=7404&top_left_x=615)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=330&width=1605&top_left_y=7866&top_left_x=307)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=463&width=1979&top_left_y=8217&top_left_x=87&polygon=1341,1,1319,23,1187,23,1165,45,682,45,660,67,309,67,287,89,221,89,199,67,133,67,111,89,67,89,23,133,1,133,1,177,89,264,111,264,177,330,133,374,133,396,111,418,67,418,45,440,45,462,199,462,199,440,177,440,133,396,155,374,155,352,199,308,199,286,265,221,287,221,309,242,506,242,528,221,770,221,792,199,1934,199,1956,177,1956,155,1978,133,1978,45,1934,1,1627,1,1605,23,1539,23,1517,1,1473,1,1451,23,1363,23) D - scortant
$\_\_\_\_$
$\_\_\_\_$ Tuile $\int_{0,0} f d_{0}=\ln \iint_{0}\left(d_{0}\right)>0$
$\Delta_{10} \Delta_{10}+\Delta_{2}+\psi_{11}$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=265&width=1451&top_left_y=9975&top_left_x=0) $\_\_\_\_$ ) calmost areajo spended
$\_\_\_\_$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=242&width=2287&top_left_y=10371&top_left_x=43) $\int|\bar{f}| d_{p}\left|\leq \int\right| \bar{f} \mid d_{p}$ and $\frac{S(E)}{\lambda} \rightarrow \bar{f}$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=243&width=2088&top_left_y=10942&top_left_x=0) I whand, $a^{-1}$ S.Cf. T $=\left[\frac{S_{n+1}(E)-f}{n}\right] =\left(\frac{N+1}{N}\right) \frac{S_{n-}(C)}{N+1}-\frac{f}{N}$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=264&width=946&top_left_y=11997&top_left_x=65) $D_{a, b}=\left\{\omega \in \Omega: \lim _{n \rightarrow \infty} n\right\}$ 'S. Cf $c a<b<\operatorname{low}_{n \rightarrow \infty} n^{-1} b_{(x)}$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=220&width=2154&top_left_y=12524&top_left_x=43&polygon=1209,0,2132,22,2154,66,2132,198,1714,220,374,220,0,198,0,22,22,0) For show bef s.t. $\mu(\theta)<\infty$ sed $y=f-b z_{\theta}$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=287&width=1693&top_left_y=12963&top_left_x=263)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=221&width=1847&top_left_y=13271&top_left_x=197&polygon=45,0,23,22,23,44,1,66,1,154,23,176,23,198,484,198,506,220,616,220,638,198,660,198,682,220,1319,220,1341,198,1363,198,1385,220,1407,220,1429,198,1451,198,1473,220,1495,220,1517,198,1824,198,1846,176,1846,44,1781,44,1759,22,1737,22,1715,0,1693,0,1671,22,1209,22,1187,44,462,44,440,22,67,22) $=\int_{0,4}\left(f \cdot v q_{0}\right) q_{p}-\int_{0} f\left(q_{p}-6 \mu(t)\right.$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=287&width=1781&top_left_y=13930&top_left_x=43) $\therefore \mu\left(O_{a, k}\right)<\infty -\alpha\left(0_{0,1}\right) \leq \int_{0}-f d \mu b_{\mu}\left(a_{a_{1}}\right) \leq \int_{a_{2}} f a_{k} \leq a_{\mu}\left(a_{0}, a_{1}\right) \therefore \mu\left(O_{0,1}\right)=0$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=243&width=2155&top_left_y=15644&top_left_x=21) $E: \bigcup_{M}, \theta_{Q}, N \rightarrow \mu(E)=0$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=221&width=1473&top_left_y=16391&top_left_x=197) artine $\bar{F}=5 \frac{M}{M} \frac{B(t)}{n}$ ,$\omega \in \epsilon^{c}$ Forgy, $\int\left|6 \cdot r^{-1}\right| d_{p}-\int|f| d_{p}$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=243&width=1385&top_left_y=17182&top_left_x=395) by Fato, $\int\left[i f \mid d x-\rho_{n \rightarrow \infty} \min _{n \rightarrow \infty}\left(n^{-1} \sqrt{n}(t)\right) d d_{n}\right. \lim _{n \rightarrow \infty} \int \ln _{n}^{-1} \operatorname{sen}_{n}(E) d \sin _{n} \int 164 \alpha$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=242&width=1779&top_left_y=18544&top_left_x=0&polygon=1757,0,1779,22,1779,176,1735,220,1669,242,527,220,43,198,21,176,0,110,0,22,43,0) Let $\mu(\Omega)<\infty$ and $p<L_{1}(\infty)=$ Than, $\forall \in C_{1}^{*}\left(R_{1}^{*}\right.$ F. $\left.N\right)$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=243&width=2309&top_left_y=18896&top_left_x=43)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=242&width=1781&top_left_y=19160&top_left_x=43)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=221&width=2309&top_left_y=19599&top_left_x=43) $g=\min [\max \{-c, c\}, c]$ by endeff's Themem $\frac{S_{n}(s)}{n} \rightarrow \bar{J}$ We that $\left|n^{-1}\right| s_{1}(s) \mid \leqslant c$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=507&width=946&top_left_y=20170&top_left_x=1296)
$\vec{n}-15,(y)-\vec{y} \|_{p}<\frac{2}{3}$ - $4 \bar{f}-\bar{g} \mid$ - y $\|_{p}^{e} =\int \max _{n \rightarrow \infty}\left|n^{-1} \operatorname{Sin}_{n}(t-s)\right|^{d} d \mu$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=264&width=1077&top_left_y=21665&top_left_x=571) $\|f=9\|_{r}^{i} \ll \frac{2}{3}$ for $n>\mathrm{N} \left\|\frac{H_{s}(t)}{n}-\bar{f} U_{p} \leq\right\| \frac{S_{s}}{n}(f-g)\left\|_{p}+\right\| \frac{S_{n}}{n}(g)-\bar{J}\left\|_{p}+\right\| \bar{F}-\overline{H_{p}}$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=332&width=2334&top_left_y=22741&top_left_x=20)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=287&width=2313&top_left_y=23092&top_left_x=41) tF ,.. $\quad d F(A)=\int \frac{1}{A} \mu \mathrm{~F}$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=352&width=2177&top_left_y=23840&top_left_x=43) $\rho=\sigma(N)$ were $\left.A=\sum_{n=1}^{\pi_{n}}: A_{n} \in B \forall_{n}, A_{n}=\mathbb{R}\right\}$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=395&width=2286&top_left_y=24719&top_left_x=65&polygon=352,0,1231,22,2264,176,2286,197,2286,351,2242,395,22,395,0,373,0,22,44,0)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=331&width=2143&top_left_y=25159&top_left_x=38)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=286&width=1496&top_left_y=25532&top_left_x=43)

Theam (sclew)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=335&width=2142&top_left_y=26019&top_left_x=38) Prove: Prove: not $f: \mathbb{R}^{2} \sim \mathbb{R}$
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=352&width=2045&top_left_y=27004&top_left_x=87)
![](https://cdn.mathpix.com/cropped/29c59e08-e494-45b8-8250-1c6bdf808a29-9.jpg?height=287&width=1386&top_left_y=27333&top_left_x=43) $\int F d x=\ln _{n \times m}\left(\int_{n}-\int_{n}(6) d x=\int f d x\right.$