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%  Lecture 8
%  Source: handwritten notes (Mathpix-converted) + Kashlak STAT 571
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\section[Lecture 8 -- \texorpdfstring{$L^p$}{Lp} Spaces and Classical Inequalities]{Lecture 8 \textemdash{} \texorpdfstring{$L^p$}{Lp} Spaces and Classical Inequalities}
\label{sec:lec08}

With integration in hand we now organise integrable functions into the
Banach spaces $L^{p}(\Omega,\Fcal,\mu)$ and collect the classical
inequalities on which the rest of probability theory rests:
Markov/Chebyshev/Chernoff (tail bounds from moments), Jensen
(convexity moves outside the integral), and H\"older/Minkowski (the
duality and triangle inequality that make $\|\cdot\|_{p}$ a norm).

\subsection{\texorpdfstring{$L^{p}$}{Lp} spaces}

Throughout, $(\Omega,\Fcal,\mu)$ is a measure space and all functions
$f,g\colon\Omega\to[-\infty,\infty]$ are measurable.

\begin{definition}{$L^{p}$ space}{lp-space}
For $1\le p<\infty$, write $f\in L^{p}(\Omega,\Fcal,\mu)$ if
\[
  \int |f|^{p}\,d\mu \;<\; \infty.
\]
For $p=\infty$, write $f\in L^{\infty}(\Omega,\Fcal,\mu)$ if
\[
  \inf\{\,t\in[-\infty,\infty] \,:\, |f|\le t\ \mu\text{-a.e.}\,\} \;<\; \infty.
\]
The associated \emph{$L^{p}$ norms} are
\[
  \|f\|_{p}
  \;=\;\left[\int |f|^{p}\,d\mu\right]^{1/p}\quad(1\le p<\infty),
  \qquad
  \|f\|_{\infty}
  \;=\;\esssup |f|
  \;=\;\inf\{\,t\,:\,\mu(\{|f|>t\})=0\,\}.
\]
\end{definition}

\begin{remark}
$L^{p}$ spaces are the canonical example of Banach spaces (complete
normed linear spaces). Subadditivity of $\|\cdot\|_{p}$ is exactly
Minkowski's inequality (\cref{thm:minkowski}); the other norm axioms
require quotienting by the relation ``$f=g$ \,$\mu$-a.e.''
\end{remark}

\begin{definition}{Conjugate indices}{conjugate-indices}
Indices $p,q\in[1,\infty]$ are \emph{conjugate} if
\[
  \frac{1}{p}+\frac{1}{q} \;=\; 1,
\]
with the convention $1/\infty=0$ so that $1$ and $\infty$ are
conjugates.
\end{definition}

\subsection{Markov, Chebyshev and Chernoff}

The next theorem bounds the measure of a level set by a single
integral. In probability it controls a tail probability by a moment.

\begin{theorem}{Markov's Inequality}{markov}
Let $f\ge 0$ be measurable and $t>0$. Writing
$\{f>t\}:=\{\omega\in\Omega:f(\omega)>t\}$,
\[
  \mu(\{f>t\}) \;\le\; t^{-1}\!\int f\,d\mu.
\]
\end{theorem}

\begin{corollary}{Chebyshev's Inequality}{chebyshev}
Let $f$ be measurable and $m\in\R$. Then
\[
  \mu(\{|f-m|>t\}) \;\le\; t^{-2}\!\int (f-m)^{2}\,d\mu.
\]
For a probability measure and a random variable $X$ with mean $\E X$,
this specialises to
\[
  \P(|X-\E X|>t) \;\le\; \Var(X)/t^{2}.
\]
\end{corollary}

\begin{corollary}{Chernoff's Inequality}{chernoff}
Let $f$ be measurable and $\eta\in\R$. Then
\[
  \mu(\{f>t\}) \;\le\; e^{-\eta t}\!\int e^{\eta f}\,d\mu.
\]
For a probability measure and random variable $X$, the right-hand
integral is the moment generating function
$M_{X}(\eta)=\E[e^{\eta X}]$ (Laplace transform); optimising over
$\eta$ yields exponential tail bounds.
\end{corollary}

\begin{remark}
All three bounds follow by applying monotonicity of the integral to
the pointwise inequality $t\,\indic_{\{f>t\}}\le f$ (or its variants
$\indic_{\{|f-m|>t\}}\le t^{-2}(f-m)^{2}$ and
$\indic_{\{f>t\}}\le e^{-\eta t}e^{\eta f}$). The names ``Markov'',
``Chebyshev'' and ``Chernoff'' are used inconsistently in the
literature; we adopt Kashlak's convention.
\end{remark}

\subsection{Jensen's inequality}

\begin{definition}{Convex function}{convex}
Let $I\subseteq\R$ be an interval. A function $\varphi\colon I\to\R$
is \emph{convex} if for all $t\in[0,1]$ and $x,y\in I$,
\[
  \varphi(tx+(1-t)y) \;\le\; t\,\varphi(x)+(1-t)\,\varphi(y).
\]
\end{definition}

\begin{theorem}{Jensen's Inequality}{jensen}
Let $(\Omega,\Fcal,\P)$ be a probability space and $X$ an integrable
random variable with $X\colon\Omega\to I\subseteq\R$. For any convex
$\varphi\colon I\to\R$,
\[
  \varphi\!\left(\int X\,d\P\right) \;\le\; \int \varphi(X)\,d\P,
  \qquad\text{i.e.}\qquad
  \varphi(\E[X]) \;\le\; \E[\varphi(X)].
\]
\end{theorem}

\begin{remark}
The probability hypothesis is essential: the proof picks a
supporting line $\varphi(x)\ge ax+b$ at $x=\E X\in I^{\circ}$ and
takes expectations, which only collapses the constant term when
$\P(\Omega)=1$.
\end{remark}

\subsection{H\"older and Minkowski}

H\"older's inequality is the dual pairing of $L^{p}$ with $L^{q}$;
Minkowski is the triangle inequality that promotes $\|\cdot\|_{p}$ to
a genuine norm.

\begin{theorem}{H\"older's Inequality}{holder}
Let $p,q\in[1,\infty]$ be conjugate indices and $f,g$ measurable.
Then
\[
  \|fg\|_{1} \;\le\; \|f\|_{p}\,\|g\|_{q}.
\]
\end{theorem}

\begin{corollary}{Cauchy--Schwarz Inequality}{cauchy-schwarz}
For measurable $f,g$,
\[
  \|fg\|_{1} \;\le\; \|f\|_{2}\,\|g\|_{2}.
\]
This is the special case $p=q=2$ of \cref{thm:holder}.
\end{corollary}

\begin{theorem}{Minkowski's Inequality}{minkowski}
Let $p\in[1,\infty]$ and $f,g$ measurable. Then
\[
  \|f+g\|_{p} \;\le\; \|f\|_{p}+\|g\|_{p}.
\]
In particular $L^{p}$ is closed under sums and $\|\cdot\|_{p}$ is
subadditive.
\end{theorem}

\subsection{Approximation by simple functions}

An immediate payoff of Minkowski's inequality is that the simple
functions are dense in $L^{p}$, mirroring their role in defining the
integral itself.

\begin{theorem}{$L^{p}$ approximation}{lp-approx}
Let $(\Omega,\Fcal,\mu)$ be a measure space and $\Acal$ a $\pi$-system
with $\sigma(\Acal)=\Fcal$, $\mu(A)<\infty$ for every $A\in\Acal$, and
$A_{i}\uparrow\Omega$ for some $A_{i}\in\Acal$. Set
\[
  V_{0} \;:=\; \left\{\,\sum_{i=1}^{n} a_{i}\,\indic_{A_{i}}
  \,:\, a_{i}\in\R,\ A_{i}\in\Acal,\ n\in\N\,\right\}.
\]
For $p\in[1,\infty)$, $V_{0}\subset L^{p}$, and for every $f\in L^{p}$
and $\varepsilon>0$ there is $v\in V_{0}$ with
$\|f-v\|_{p}<\varepsilon$.
\end{theorem}

\begin{remark}
A standard instance: $(\R,\Bcal,\lambda)$ with $\Acal$ the
$\pi$-system of half-open bounded intervals
(see~\cref{prop:halfopen-ring}) gives density of step functions in
$L^{p}(\R)$.
\end{remark}
