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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}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{29c59e08-e494-45b8-8250-1c6bdf808a29-1_643_800_2445_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{center}
\end{figure}

\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-1_1819_2352_3714_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

$$f f d_{\mu}=\sum_{i x, \mu(B i)}^{a}$$

\begin{center}
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-1_3058_2367_4576_0}
\end{center}

$$\begin{aligned}
& \text { Secordly, }(Y(, Y) \text { is otten }(\mathbb{R}, B(\mathbb{R})) \\
& \text { ( } \left.\mathbb{R}, \beta(\mathbb{R})^{+}\right) \\
& \text {In this ase, me a say that } f \text { is Borel measurable } \\
& \text { Similarly, me can replace } \beta(\mathbb{R}) \text { with } M_{\lambda}(\mathbb{R}) \text { to } \\
& \text { Set lebesgue measurable finctions. }
\end{aligned}$$

\begin{center}
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-1_157_1301_8902_62}
\end{center}

\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\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-1_267_1975_12131_78}\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-1_236_2348_12570_31} a courtable pacrsection of messivelle sits is restivelse\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-1_189_2285_12931_94} $=$ " $\left.\left\{x: f^{(x)}\right\}+1\right\}$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-1_205_1129_13448_78}\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-1_266_1156_13655_87}\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-1_189_2227_13903_125}\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-1_204_2335_14075_31}\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-1_205_2274_14294_31}\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-1_220_2273_14451_31} over\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-1_330_2336_15047_31}\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-1_283_2321_15235_31} West wer $\sigma\left(\Sigma f^{-1}(\omega): u \subset R\right.$, ven $\left.\xi\right) \leq \beta(x)$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-1_235_2147_15580_31}\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-1_266_2132_15674_31}\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-1_235_2317_15862_62}\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-1_249_2171_16011_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 $N=\left\{N: f(u) \neq g^{(L)}\right\} \mu(N)=0$\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-1_267_2007_17367_141} for awth probablify 7\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-1_205_2242_17711_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.\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-1_424_550_18605_1786}\\
that is, $\lambda\left(\mathrm{C}_{0}, 179 \mathrm{Q}\right)=0$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-1_314_2211_19326_94}\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-1_221_2211_19545_62} $m_{1}$ and notert $\left(\tau_{m}=\Sigma 2^{-m}, q_{m}+\Sigma 2^{-m+1}\right)$

Lecture 6: Infegertion \& Gonvergerve Theorens\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-2_247_1482_0_403}\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-2_247_1728_246_358}\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-2_293_1707_448_358}\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-2_270_1415_785_358}\\
Notation. I 1\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-2_405_1751_1121_314}\\
$\_\_\_\_$

\section*{\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-2_696_1998_2109_44} $\therefore A \subset \mathcal{C} \Rightarrow I=$ T \includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-2_426_2266_2961_44} \includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-2_360_853_3410_269} $\therefore 6$ or $+9=$ \\
 \includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-2_292_1190_4173_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 $f(\omega) \leq 0$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-2_10749_2379_5339_0}\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-2_2491_2379_16289_0}\\
and $g^{*} \uparrow f$ so one again $\rho_{g^{*}} d_{\mu} \uparrow \int f d_{k}$ out $g^{*} \leq f_{i}$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-2_338_2245_19654_44}\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-2_203_1864_19968_44}\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-2_292_944_20148_44}\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-2_338_1302_20372_67} ∴ form and\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-2_315_1370_20686_201}\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-2_270_1549_20888_89} b. a good $f=f^{+} f^{+}$, we have $f^{+} \uparrow f^{+}$and folf\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-2_270_1100_21404_67} ∴ From concre $\int f_{i}^{-} d_{k} d \int f^{-} d x \because \quad \sin f_{i} d_{\mu} \upharpoonleft \int f d_{\mu}$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-2_315_1258_22302_44}\\
Theorem (Fifery leammal):\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-2_292_2042_22885_67} Then, $\rho_{\text {limuin }} f_{i} d_{\mu} \leq$ limini $\int f_{i} d \mu$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-2_293_1550_23603_44}\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-2_360_1549_23917_89} Then s $_{j} \uparrow$ miniff and $f_{i} \geq 0$\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-2_383_1975_24388_112} $\therefore \quad \int_{j_{j}} d_{\mu} \leq \int f_{i} \alpha_{\mu} \theta_{i}$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-2_695_2311_25129_44}\\
Theorem (conoputed amergenac)\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-2_360_2178_25936_44} If $\left|f_{i}\right| \leq g \quad \forall_{i}$ and $f(b) \rightarrow f(c)$ to ench $w \in \Omega$ then\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-2_248_1998_26654_44} Then $f_{1}^{1} \leq f_{i} \leq f_{i}^{2} \quad$ Not them $\int f_{i}^{\wedge} d \mu \uparrow \int f_{d p}$\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-2_336_2221_28001_89} $f_{1}^{\wedge} d_{\mu} \leq \int f_{1} d \mu \leq \int f_{1}{ }^{v} d \mu$\\
$\int_{f f_{\mu} \leq} \leq \int f_{\mu} \leq \int f d_{\mu} \leq$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-3_238_1664_0_438}

\begin{center}
\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}
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-3_148_162_3515_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}
\end{center}

\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 $k \epsilon \beta$}
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-3_238_2194_8626_18}
\end{center}

\begin{center}
\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}
\end{center}

Theorem chonotine class Theoren\\
Let $A$ be a field a m be a monstane closes s.t. $A<m$, then $f(A) \leq M$.\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-3_7142_2379_16246_0}\\
and $\iint \Sigma_{\varepsilon}\left(x_{1}\right) d \alpha_{1}(\mathcal{G}) d \mu A \in F$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-4_379_1078_7076_757}

Proof C For ve\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-4_351_1603_7862_378} $T(\varepsilon)=\iint_{E}(x, y) d_{\mu}(x) d x(y)$ any $\varepsilon_{0} \cdot x \times y$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-4_438_1923_8561_58}

Whomsh: st. $\rho(A \times B)=\mu(A) \gamma(B)$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-4_380_1661_9755_87}\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-4_351_1399_10250_58}\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-4_409_1166_10774_87}\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-4_613_1078_10716_1223} -, $\mu\left(N_{i}\right)<\infty \quad v\left(B_{i}\right)<\infty$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-4_350_991_11881_87}\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-4_467_962_12259_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}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{mide 6}
  \includegraphics[alt={},max width=\textwidth]{29c59e08-e494-45b8-8250-1c6bdf808a29-4_321_350_17356_1485}
\end{center}
\end{figure}

ol f-

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{mide 6}
  \includegraphics[alt={},max width=\textwidth]{29c59e08-e494-45b8-8250-1c6bdf808a29-4_380_321_17297_2009}
\end{center}
\end{figure}

$\quad \int f^{+}(x, y) d \mu^{(x)}<\infty$\\
$\quad f^{-}(x, y) d y(y)<\infty \quad \mu$.\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-4_437_2098_17822_174} $\int f(x, y) d v(y)=\int f^{+}(x, y) d v(y)-\int f(x, y) d v(y), \ldots, c \iint f(x, y) d v(y) d \mu(x)=\iint f^{+}(x, y) d v(y) d \mu(x)$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-4_584_2127_18899_58}\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-4_438_1486_19540_29} spaces Then, $\Omega=\Omega_{n}$\\
$w_{i=1}^{\infty} \Omega_{i=1}^{\infty}$ s.t. $w_{i} \in \Omega_{n}$\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-4_612_2243_20268_58} for $A_{n} \in \mathcal{F}_{n}$ then $R=\square \mathcal{F}_{n} A_{n}$ Dearte $S$ to be the fudg genertad by $P$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-4_437_2243_21433_58}\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-4_379_1748_21695_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 $\underline{A}_{x}=\{y \in y:(x, y) \in A\}$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-4_13163_2301_24083_0}\\
$\|f\|_{\infty}$\\
$=\inf \{f \in[-\infty, \infty]: \mu(\{|f|>+\})=0\}$

\begin{itemize}
  \item ess spe \{lfl\}\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-5_598_664_2540_1436}\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-5_222_796_2872_0}\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-5_288_1392_3225_44}\\
, 1 and $\infty$ an anyouges.\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-5_287_1966_3712_0}\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-5_244_1105_3910_0}\\
ten $\{f\}<\}=\{\omega \in \Omega: f(\omega)>i\}$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-5_266_1481_4352_265}\\
$\mu(2 f f+3) \leqslant \frac{1}{4} \int f d \mu$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-5_266_1238_4905_44}\\
$t_{p}(2 \leqslant z+3)=+\int \eta_{p+x} d x$ $\_\_\_\_$\\
Ant\\
$\leq \int f d \mu$\\
$\Rightarrow \mu(\varepsilon f+t) \leq \frac{1}{+} \int f d \mu$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-5_266_1503_5965_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, $\eta \in \mathbb{R}$\\
$\quad \mu(\varepsilon f+3) \leq e^{-\eta+} \int e^{\eta f} d \mu$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-5_399_1216_7821_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)\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-5_421_2232_9191_0}\\
have $\quad \phi\left(\int x d r\right) \leq \int \phi(x) d r$\\
T.e. $\phi(\mathbb{E} x) \leq \mathbb{E}[\phi(x)]$
\end{itemize}

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 $\neq \infty$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-5_288_885_11622_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 $\| f l l_{p}=1$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-5_376_1393_13301_110}\\
$\square \mathbb{P}(A)=\int_{A} f d \mu$\\
Then, wia Tersen and moting thit $2(p-1)=p$\\
$\|f g\|_{1}=\int(f g) d \mu=\int \frac{1 g \mid}{f\left(r^{-1}\right.} \frac{1}{f(f \mid>0}|f|^{p} d \mu$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-5_354_1636_14185_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)\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-5_398_2299_16505_44}\\
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\\
$\left\|\left.f_{f+g}\right|^{r^{-1}}\right\|_{q}=\left[\int\left(f_{f}\right]^{(f-)} q_{q}\right]^{\frac{1}{2}}$\\
$\left[\int|i+g|^{p} d \mu\right] \frac{t-1}{r}$\\
$\left\|f^{+} J\right\|_{p}^{+-1}$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-5_244_2122_19311_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, $A \Omega$ :\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-5_309_2320_20659_44} \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 $\forall f \in L^{p}, \forall z>0 \quad \exists \in V_{0}$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-5_377_2321_21520_44}\\
$\mu(A)^{\frac{1}{c}}<\Delta$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-5_421_2321_21984_44}\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-5_465_2216_22565_31}\\
$\left\|(f+g)-\left(v_{f}+v_{y}\right)\right\|_{p} \leq\left\|f-v_{f}\right\|_{p}+\left\|_{q}-v_{y}\right\|_{p}<2 \varepsilon$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-5_575_1769_23465_88}\\
We have $A \subset \mathcal{L}$ ad ∴ $\Omega \in \mathcal{Z}{ }_{\text {for }}$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-5_266_1305_24260_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 $\left[n, i^{-n}\right]$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-5_355_2321_26138_44}\\
and $\mid f-$ m $\left._{n}\right|^{p} \leq|f|^{p} \leq$ bound

\begin{itemize}
  \item By Domint-al convegerres $\left(f-f_{n} \| p-\left[\int\left|f-f_{n}\right|^{p} d \mu\right]^{\bar{P}} \rightarrow 0\right.$
\end{itemize}

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{center}
\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}
\end{center}

\begin{center}
\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}
\end{center}

\begin{center}
\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}
\end{center}

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\\
$\therefore \liminf A_{i}=\bigcup_{j=1}^{\infty} a_{j>i} A_{j}$\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-6_263_2289_17849_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 $\left.A_{i}\right)=0$\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-6_228_2271_18652_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{center}
\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}
\end{center}

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 $(\mathbb{R}, B)$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-7_169_812_338_33}

\begin{itemize}
  \item $\mathbb{P}(x \in A)=\mathbb{P}\left(\sum \omega G \Omega: x(\omega) \in A()\right.$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-7_440_1454_743_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 $(y, y)$.
\end{itemize}

\begin{center}
\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\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-7_427_657_5415_1618}
 \\
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Proot: $w \log c=0$, therwise replace $x_{i}$ with $x_{i}=c$ Then for any $t>0$, Chelysher says, \\
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 \\
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\(\mathbb{P}\left(\frac{\left|S_{n}\right|}{n} \geqslant t\right) \leq \frac{\mathbb{E} S_{n}^{2}}{t^{2} n^{2}}=\sum_{1, j=1}^{n} \frac{\mathbb{E}\left[x_{i} x_{j}\right]}{t^{2} n^{2}}=\frac{1}{n t^{2}} \rightarrow 0\) \\
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In the rext theorm, me require independent $X_{1}$ 's' me also need the voriance \(V_{o r}(x)=\int(x-\mathbb{E} x)^{2} d \mathbb{P}(\omega)\) \\
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Thomem (Strong Loge Nonbers) \\
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het $\left\{x_{i}\right\}_{i}^{\infty}=$ rentom veriables from $\Omega \rightarrow \mathbb{R}$, \\
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\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\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-7_299_351_8004_1992}
 \\
finite limit. ${ }^{(2)}$ \\
\end{tabular} \\
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Proof: (2) Assume $n^{-1} S_{n} \rightarrow C \in \mathbb{R}$ att afo $\mathbb{E}\left|X_{1}\right|=\infty$ Note that $\frac{x_{n}}{n}=\frac{S_{n}-S_{n-1}}{n} \rightarrow 0$ \(\frac{S_{n}}{n}-\frac{S_{n-1}}{n} \rightarrow c-c=0\) \\
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Since $\mathbb{E}\left|x_{i}\right|=\infty$ then $\sum_{n=0}^{\infty} \mathbb{P}\left(x_{i}>n\right)=\infty$, twe Borel - cartell: \\
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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 \(\therefore n^{-1} S_{n}+c \Rightarrow \mathbb{R}\)\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-7_325_487_9699_1822}
 \\
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(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 $x^{+}, x^{-}$ \\
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Also, independence of $x, y$ implies Independence of $x^{t}$ and $\varphi^{t}$ Also, $F$ devotes the low of $X$ i.e $F(x)=\mathbb{P}(X \leq x)$ \\
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\(\begin{aligned} & \sum_{n=1}^{\infty} k_{n}^{-2} \mathbb{1}_{k n \geq a} \leq 4 \sum_{n=1}^{\infty} \delta^{-2 n} \mathbb{1}_{\delta^{n} \geq i} \leq \frac{4}{2^{2}\left(1-f^{-2}\right)} \leq c_{0} i^{-2} c^{-2} \\ & \text { for Some } c_{0}>0 \end{aligned}\) \\
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\end{tabular}
\end{center}

$\sum_{i=k+1}^{\infty} i^{2}<\int_{k}^{-1} x^{-2} d x=\frac{1}{k}$\\
Now, for any $t>0, \exists c>0$ depereling ony on $t$ and $\delta$ s.t.\\
$\sum_{n=1}^{\infty} \mathbb{P}\left(\left|T_{k_{n}}-\mathbb{E} T_{k_{n}}\right|>+k_{n}\right) \leq c_{1} \sum_{n=1}^{\infty} k_{n}^{-2} \operatorname{Var}\left(T_{k_{n}}\right)$\\
$=c_{1} \sum_{n=1}^{\infty} \frac{1}{k_{n}^{2}} \sum_{i=1}^{k_{n}} V_{r}\left(Y_{i}\right)$\\
$=c_{i}^{\infty} \sum_{i=1}^{\infty}\left[\begin{array}{ll}V_{i}-\left(y_{i}\right) & \sum_{k_{n} ?} \\ k_{n}^{2}\end{array}\right]$\\
$\leq c_{2} \sum_{1}^{\infty} c^{-2} \mathbb{E} Y_{1}^{2}$\\
$=c_{i} \sum_{i=1}^{\infty}-2 \int_{0}^{\infty} x^{2} d F(x)$\\
$\left.=c_{2} \sum_{i=1}^{n} \sum_{n=0}^{i-1} \int_{k}^{k+1} x^{2} d F(x)\right\}$\\
$\leq c_{3} \sum_{k=0}^{\infty} \frac{1}{k+1} \int_{k}^{k+1} x^{2} d F(x)$\\
$\leq c_{3} \sum_{k=0}^{n} \int_{k}^{k+1} x d F(x)$\\
$\leq c_{3} \sum_{k=0}^{\infty} e^{-} x_{i}<\infty$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-7_423_863_15379_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{center}
\begin{tabular}{|l|}
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\multirow{2}{*}{\begin{tabular}{l}
Prohorov's Theorem "Heren bout congaretress" \\
Defn : (Onform Tghtress) \\
\end{tabular}} \\
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\multirow[b]{2}{*}{} \\
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 \\
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uniformy tight if $\forall \varepsilon>0$ JK compart s.t. $\mu_{i}\left(K_{\varepsilon}\right)>1-\varepsilon$ \\
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Theorem (Proborous) $\leftarrow$ couple different versions \\
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$\Rightarrow \mu$ to some $\mu$ depe \\
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Apposition: \\
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If $\left\{\mu_{i}\right\}_{j=1}^{\infty}$ and $\left(\mu\right.$ ore prob reasures st. $\theta_{\text {ik }} \mu_{i_{k}}=\mu$ \\
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\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}} \\
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 \\
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Defn (Gonssi- measire) \\
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A Broel measure $\gamma$ on $(\mathbb{R}, T B)$ id said to be Cearston with mean $m$ and voriance $g^{2}$ if \\
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$\gamma((a, b])=\frac{1}{\sigma \sqrt{2 \pi}} \int_{a}^{b} \exp \left[-\frac{1}{2 \sigma^{2}}(x-m)^{2}\right] d \lambda(x)$ \\
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\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}} \\
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 \\
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 \\
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$\gamma$ 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 \\
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Defn ( $R . V$ ) \\
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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. \\
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For rectors $u, v \in \mathbb{R}^{\alpha}$, the inver product $\langle u, v\rangle=\left\{4 v_{i}\right.$ \\
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\multirow{2}{*}{\begin{tabular}{l}
$141^{2}=\langle 4,4\rangle$ \\
Chereforsistic foretion \\
\end{tabular}} \\
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 \\
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 \\
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\(\begin{gathered} \tilde{\mu}(\nu):=\int \exp \{i<x, \nu>\} d \mu(x) \\ \tilde{r}(\text { to sealling } \\ =\sqrt{1} \end{gathered}\) \\
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 \\
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 \\
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$p(x)$ is the probability density funtion for $\gamma$ a.e. \\
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\begin{tabular}{l}
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-8_136_698_7900_1621}
 \\
Convolution fintions" \\
\end{tabular} \\
\hline
\multirow{3}{*}{\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-8_598_2308_8148_56}
} \\
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 \\
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 \\
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 \\
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Also, for indelpendent $C V^{\prime} \backslash X, Y$ with megures $\mu_{\nu} \nu$ X $X Y$ has reduced measure $\mu * \nu$ \\
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Theoren (uniqueness of $\sim$ ) \\
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Proof : Let $\gamma_{j}$ be a mean zero Gairsia measure a $\mathbb{R}^{d}$ with co-vertance $\sigma^{2} \mathbb{P}_{d}$. \\
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Devote $\mu^{(\sigma)}=\mu * \gamma_{0}$ and Same for $\nu^{(\sigma)}$. \\
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It cal be shown that the correspondiry sensity functions for $\mu^{(\sigma)}$ and $\nu^{(\sigma)}$ are \\
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\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}} \\
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\multirow[t]{2}{*}{} \\
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\multirow[b]{2}{*}{} \\
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 \\
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Let $S_{n}=\sum_{j=1}^{n} x_{j}$, then $n^{-\frac{1}{2}} S_{n} \rightarrow z$, where $z$ is Caussion \\
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 \\
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\end{tabular}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-8_1658_2338_14547_0}
\end{center}

\begin{center}
\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}}} \\
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 &  &  \\
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 &  &  \\
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\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" $=\mathbb{E}\left|x_{j}\right|^{2}$} \\
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\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
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-8_210_1118_18001_41}
 &  &  \\
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\(\mathbb{P}\left(\left|n^{\frac{-1}{2}} \Sigma_{n}\right|>n_{\varepsilon}\right)<\varepsilon\) & \includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-8_224_545_18196_1688}
 & \includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-8_266_545_18168_1674}
 \\
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\multicolumn{3}{|l|}{\(\therefore \text { Seq } n^{\frac{-1}{2}} S_{n} \mapsto \quad u .7\)} \\
\hline
\multicolumn{3}{|l|}{\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-8_168_1969_18782_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|}{$\sqrt{ }$} \\
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$h(0)=1 \nabla L(0)=0 \nabla^{2} h(0)=-f\left[x_{j} x_{j}^{\top}\right]$ \(\text { foren: } B^{2}\) & Geldear Nom & \includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-8_196_615_19759_1423}
 \\
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\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}
\end{center}

\section*{\begin{center}
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_308_1407_571_395}
\end{center}}
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_220_1605_747_351}\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_243_1649_966_351} resienting $x$ and $a\left(T^{-1}(A)\right)$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_264_616_1538_351}

\begin{itemize}
  \item set $A e^{\prime} z^{\prime \prime}$ is note: the vilution\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_265_572_1999_395}\\
$\_\_\_\_$ $\forall A \in F$\\
$\_\_\_\_$\\
$f=f \cdot T$\\
$=f(T \cdots)$ f vani (Cengodite)\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_243_770_2724_373} Garres es $(0,1], B, \lambda)$ Subt mas : $T(x)=x+a$ mod $\begin{cases}x+a & x \\ x+a-1, & x+a>1\end{cases}$\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_265_1539_3581_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 $\_\_\_\_$\\
$\_\_\_\_$\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_309_418_4350_1516} Engotic Thorems $\Omega, F$,\\
$\_\_\_\_$\\
$\_\_\_\_$ not me obefiee $S_{n}=S_{n}(f)=f+f$
\end{itemize}

Changent\\
$\_\_\_\_$\\
$\_\_\_\_$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_397_1100_6437_21}\\
$\_\_\_\_$\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_242_1824_6899_43} Samemo a\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_287_1231_7404_615}\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_330_1605_7866_307}\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_463_1979_8217_87} D - scortant\\
$\_\_\_\_$\\
$\_\_\_\_$ Tuile $\int_{0,0} f d_{0}=\ln \iint_{0}\left(d_{0}\right)>0$\\
$\Delta_{10} \Delta_{10}+\Delta_{2}+\psi_{11}$\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_265_1451_9975_0} $\_\_\_\_$ ) calmost areajo spended\\
$\_\_\_\_$\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_242_2287_10371_43} $\int|\bar{f}| d_{p}\left|\leq \int\right| \bar{f} \mid d_{p}$ and $\frac{S(E)}{\lambda} \rightarrow \bar{f}$\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_243_2088_10942_0} I whand, $a^{-1}$ \href{http://S.Cf}{S.Cf}. T $=\left[\frac{S_{n+1}(E)-f}{n}\right] =\left(\frac{N+1}{N}\right) \frac{S_{n-}(C)}{N+1}-\frac{f}{N}$\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_264_946_11997_65} $D_{a, b}=\left\{\omega \in \Omega: \lim _{n \rightarrow \infty} n\right\}$ 'S. Cf $c a<b<\operatorname{low}_{n \rightarrow \infty} n^{-1} b_{(x)}$\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_220_2154_12524_43} For show bef s.t. $\mu(\theta)<\infty$ sed $y=f-b z_{\theta}$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_287_1693_12963_263}\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_221_1847_13271_197} $=\int_{0,4}\left(f \cdot v q_{0}\right) q_{p}-\int_{0} f\left(q_{p}-6 \mu(t)\right.$\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_287_1781_13930_43} $\therefore \mu\left(O_{a, k}\right)<\infty -\alpha\left(0_{0,1}\right) \leq \int_{0}-f d \mu b_{\mu}\left(a_{a_{1}}\right) \leq \int_{a_{2}} f a_{k} \leq a_{\mu}\left(a_{0}, a_{1}\right) \therefore \mu\left(O_{0,1}\right)=0$\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_243_2155_15644_21} $E: \bigcup_{M}, \theta_{Q}, N \rightarrow \mu(E)=0$\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_221_1473_16391_197} artine $\bar{F}=5 \frac{M}{M} \frac{B(t)}{n}$ ,$\omega \in \epsilon^{c}$ Forgy, $\int\left|6 \cdot r^{-1}\right| d_{p}-\int|f| d_{p}$\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_243_1385_17182_395} by Fato, $\int\left[i f \mid d x-\rho_{n \rightarrow \infty} \min _{n \rightarrow \infty}\left(n^{-1} \sqrt{n}(t)\right) d d_{n}\right. \lim _{n \rightarrow \infty} \int \ln _{n}^{-1} \operatorname{sen}_{n}(E) d \sin _{n} \int 164 \alpha$\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_242_1779_18544_0} Let $\mu(\Omega)<\infty$ and $p<L_{1}(\infty)=$ Than, $\forall \in C_{1}^{*}\left(R_{1}^{*}\right.$ F. $\left.N\right)$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_243_2309_18896_43}\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_242_1781_19160_43}\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_221_2309_19599_43} $g=\min [\max \{-c, c\}, c]$ by endeff's Themem $\frac{S_{n}(s)}{n} \rightarrow \bar{J}$ We that $\left|n^{-1}\right| s_{1}(s) \mid \leqslant c$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_507_946_20170_1296}\\
$\vec{n}-15,(y)-\vec{y} \|_{p}<\frac{2}{3}$ - $4 \bar{f}-\bar{g} \mid$ - y $\|_{p}^{e} =\int \max _{n \rightarrow \infty}\left|n^{-1} \operatorname{Sin}_{n}(t-s)\right|^{d} d \mu$\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_264_1077_21665_571} $\|f=9\|_{r}^{i} \ll \frac{2}{3}$ for $n>\mathrm{N} \left\|\frac{H_{s}(t)}{n}-\bar{f} U_{p} \leq\right\| \frac{S_{s}}{n}(f-g)\left\|_{p}+\right\| \frac{S_{n}}{n}(g)-\bar{J}\left\|_{p}+\right\| \bar{F}-\overline{H_{p}}$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_332_2334_22741_20}\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_287_2313_23092_41} tF ,.. $\quad d F(A)=\int \frac{1}{A} \mu \mathrm{~F}$\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_352_2177_23840_43} $\rho=\sigma(N)$ were $\left.A=\sum_{n=1}^{\pi_{n}}: A_{n} \in B \forall_{n}, A_{n}=\mathbb{R}\right\}$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_395_2286_24719_65}\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_331_2143_25159_38}\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_286_1496_25532_43}

Theam (sclew)\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_335_2142_26019_38} Prove: Prove: not $f: \mathbb{R}^{2} \sim \mathbb{R}$\\
\includegraphics[max width=\textwidth, alt={}, center]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_352_2045_27004_87}\\
\includegraphics[max width=\textwidth, alt={}]{29c59e08-e494-45b8-8250-1c6bdf808a29-9_287_1386_27333_43} $\int F d x=\ln _{n \times m}\left(\int_{n}-\int_{n}(6) d x=\int f d x\right.$


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