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%  Lecture 7
%  Source: handwritten notes (Mathpix-converted) + Kashlak STAT 571
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\section[Lecture 7 -- Lebesgue-Stieltjes Measure; Fubini-Tonelli]{Lecture 7 \textemdash{} Lebesgue--Stieltjes Measure; Fubini--Tonelli}
\label{sec:lec07}

We are halfway through, with one more lecture on integration to come. The plan today is to use measurable functions to \emph{transport} measures between spaces: a measurable map $\psi:\mathbb{X}\to\mathbb{Y}$ together with a measure $\mu$ on $\mathbb{X}$ produces a pushforward $\nu=\mu\circ\psi^{-1}$ on $\mathbb{Y}$. Specialising to $\mathbb{X}=\mathbb{Y}=\R$ and $\mu=\lambda$ recovers the family of \emph{Lebesgue--Stieltjes} measures $dF$ generated by distribution-like functions $F$. We then build the \emph{product measure} $\mu\times\nu$ rigorously via a monotone class argument, and conclude with \emph{Fubini--Tonelli}, the workhorse for swapping the order of integration.

\subsection{Image measures and Lebesgue--Stieltjes}

Given two measurable spaces $(\mathbb{X},\Xcal)$ and $(\mathbb{Y},\Ycal)$ and a measurable $\psi:\mathbb{X}\to\Ycal$, any measure $\mu$ on $\Xcal$ induces an \emph{image measure} on $\Ycal$.

\begin{definition}{Image (pushforward) measure}{image-measure}
Let $\psi:\mathbb{X}\to\mathbb{Y}$ be measurable and let $\mu$ be a measure on $(\mathbb{X},\Xcal)$. The \emph{image measure} of $\mu$ under $\psi$ is the set function
\[
  \nu(B) \;=\; \mu\bigl(\psi^{-1}(B)\bigr), \qquad B\in\Ycal,
\]
also written $\nu=\mu\circ\psi^{-1}$. It is a measure on $(\mathbb{Y},\Ycal)$.
\end{definition}

The most useful instance turns Lebesgue measure $\lambda$ on $\R$ into a one-parameter family of Borel measures indexed by a ``CDF-like'' function $F$.

\begin{theorem}{Lebesgue--Stieltjes measure}{lebesgue-stieltjes}
Let $F:\R\to\R$ be non-constant, right-continuous, and non-decreasing. Then there exists a unique measure $dF$ on $(\R,\Bcal(\R))$ such that
\[
  dF\bigl((a,b]\bigr) \;=\; F(b)-F(a) \qquad\text{for all } a<b.
\]
Concretely, with $F(\pm\infty)=\lim_{x\to\pm\infty}F(x)$, $I=(F(-\infty),F(\infty))$ and
\[
  g(y) \;=\; \inf\{x\in\R:y\le F(x)\},
\]
the function $g$ is left-continuous, non-decreasing, and Borel measurable, and $dF=\lambda\circ g^{-1}$ realises the image-measure construction of \cref{def:image-measure}.
\end{theorem}

\begin{remark}
The map $g$ is the (left-continuous) \emph{generalised inverse} or \emph{quantile function} of $F$. In probabilistic language, $F$ is a CDF, $dF$ is the law it induces on $\R$, and $g(U)$ for $U\sim\mathrm{Uniform}(0,1)$ is the standard inverse-CDF sampling recipe.
\end{remark}

\begin{example}[Cumulative distribution functions]
If $F:\R\to[0,1]$ is a CDF (so additionally $F(-\infty)=0$ and $F(\infty)=1$), then $dF$ is a probability measure on $\R$. Every Borel probability measure on $\R$ arises this way, and ``integration with respect to $dF$'' is exactly expectation under that distribution.
\end{example}

\begin{definition}{Radon measure}{radon}
Let $(\Omega,\Bcal,\mu)$ be a measure space with $\Bcal$ the Borel $\sigma$-field of a topological space $\Omega$. The measure $\mu$ is a \emph{Radon measure} if $\mu(K)<\infty$ for every compact $K\in\Bcal$.
\end{definition}

\begin{remark}
Most measures one meets in practice are Radon. Going one step beyond \cref{thm:lebesgue-stieltjes}, every non-zero Radon measure $\mu$ on $\Bcal(\R)$ can be written as $dF=\lambda\circ g^{-1}$ for the choice
\[
  F(x) \;=\; \begin{cases}\mu((0,x]), & x\ge 0,\\ -\mu((x,0]), & x<0,\end{cases}
\]
since then $F(b)-F(a)=\mu((a,b])$ for $a<b$, and uniqueness in \cref{thm:lebesgue-stieltjes} forces $\mu=dF$.
\end{remark}

\subsection{Product \texorpdfstring{$\sigma$}{sigma}-fields and product measure}

We have already met product measure briefly in \cref{def:product-measure}. We now build it rigorously, using a new analogue of Dynkin's $\pi$--$\lambda$ argument tailored to fields rather than $\pi$-systems.

\begin{definition}{Product $\sigma$-field}{product-sigma}
Given measurable spaces $(\mathbb{X},\Xcal)$ and $(\mathbb{Y},\Ycal)$, the collection of \emph{rectangles}
\[
  \mathcal{R} \;=\; \{A\times B:A\in\Xcal,\,B\in\Ycal\}
\]
generates the \emph{product $\sigma$-field} $\Xcal\times\Ycal:=\sigma(\mathcal{R})$ on $\mathbb{X}\times\mathbb{Y}$.
\end{definition}

\begin{definition}{Monotone class}{monotone-class}
A collection $\Mcal$ of subsets of $\Omega$ is a \emph{monotone class} if, for any sequence $\{A_i\}_{i=1}^{\infty}\subseteq\Mcal$,
\begin{itemize}
  \item $A_i\uparrow A=\bigcup_{i=1}^{\infty}A_i$ implies $A\in\Mcal$,
  \item $A_i\downarrow A=\bigcap_{i=1}^{\infty}A_i$ implies $A\in\Mcal$.
\end{itemize}
\end{definition}

\begin{remark}
A field that is also a monotone class is automatically a $\sigma$-field: closure under countable monotone limits, combined with finite-union and complement closure, upgrades finite operations to countable ones.
\end{remark}

\begin{theorem}{Monotone class theorem}{monotone-class-thm}
Let $\Acal$ be a field on $\Omega$ and let $\Mcal$ be a monotone class with $\Acal\subseteq\Mcal$. Then $\sigma(\Acal)\subseteq\Mcal$.
\end{theorem}

This is the field-level analogue of $\pi$--$\lambda$: the smallest monotone class containing a field $\Acal$ already contains $\sigma(\Acal)$. With it, we can promote rectangle-level identities to identities on the full product $\sigma$-field.

\begin{theorem}{Existence and uniqueness of product measure}{product-measure-existunique}
Let $(\mathbb{X},\Xcal,\mu)$ and $(\mathbb{Y},\Ycal,\nu)$ be $\sigma$-finite measure spaces, and let $\pi$ be the set function on $\mathcal{R}$ defined by
\[
  \pi(A\times B) \;=\; \mu(A)\,\nu(B), \qquad A\times B\in\mathcal{R}.
\]
Then $\pi$ extends uniquely to a measure on $(\mathbb{X}\times\mathbb{Y},\,\Xcal\times\Ycal)$, and for every $E\in\Xcal\times\Ycal$,
\[
  \pi(E) \;=\; \iint \indic_{E}(x,y)\,d\mu(x)\,d\nu(y)
         \;=\; \iint \indic_{E}(x,y)\,d\nu(y)\,d\mu(x).
\]
\end{theorem}

\begin{remark}
$\sigma$-finiteness is essential: without it the extension need not be unique. The proof handles finite $\mu,\nu$ first (using the monotone class theorem applied to the field of finite disjoint unions of rectangles) and then patches together a $\sigma$-finite partition.
\end{remark}

\begin{lemma}{Order swap for indicators}{indicator-fubini}
Let $(\mathbb{X},\Xcal,\mu)$ and $(\mathbb{Y},\Ycal,\nu)$ be finite measure spaces, and define
\[
  \Fcal \;=\; \Bigl\{E\subseteq\mathbb{X}\times\mathbb{Y}:\iint\indic_{E}(x,y)\,d\mu(x)\,d\nu(y)=\iint\indic_{E}(x,y)\,d\nu(y)\,d\mu(x)\Bigr\}.
\]
Then $\Xcal\times\Ycal\subseteq\Fcal$.
\end{lemma}

The lemma gives the indicator-function case of Fubini--Tonelli; the general case follows by the standard ladder simple $\to$ non-negative $\to$ integrable.

\subsection{The Fubini--Tonelli theorem}

\begin{theorem}{Fubini--Tonelli}{fubini-tonelli}
Let $(\mathbb{X},\Xcal,\mu)$ and $(\mathbb{Y},\Ycal,\nu)$ be $\sigma$-finite measure spaces, and let $f:\mathbb{X}\times\mathbb{Y}\to\R$ be $\Xcal\times\Ycal$-measurable. Suppose either
\begin{itemize}
  \item (\emph{Tonelli}) $f\ge 0$, or
  \item (\emph{Fubini}) $\iint|f|\,d(\mu\times\nu)<\infty$.
\end{itemize}
Then
\[
  \int f\,d(\mu\times\nu) \;=\; \iint f(x,y)\,d\mu(x)\,d\nu(y) \;=\; \iint f(x,y)\,d\nu(y)\,d\mu(x).
\]
Moreover, $x\mapsto\int f(x,y)\,d\nu(y)$ is $\Xcal$-measurable and $y\mapsto\int f(x,y)\,d\mu(x)$ is $\Ycal$-measurable.
\end{theorem}

\begin{remark}
The Tonelli half (non-negative $f$) needs only $\sigma$-finiteness; the Fubini half additionally needs absolute integrability. In practice one often runs Tonelli on $|f|$ first to verify the Fubini hypothesis, then applies Fubini to $f$ itself.
\end{remark}

\begin{example}[Sections of measurable sets]
For $A\in\Xcal\times\Ycal$ and $x\in\mathbb{X}$, the \emph{section}
\[
  A_x \;=\; \{y\in\mathbb{Y}:(x,y)\in A\}
\]
lies in $\Ycal$, and applying \cref{thm:fubini-tonelli} to $f=\indic_{A}$ gives
\[
  (\mu\times\nu)(A) \;=\; \int \nu(A_x)\,d\mu(x) \;=\; \int \mu(A^{y})\,d\nu(y),
\]
where $A^{y}=\{x:(x,y)\in A\}$. This is the measure-theoretic incarnation of ``area equals the integral of the cross-sectional length''.
\end{example}

\begin{remark}
Next lecture we round out integration: $L^{p}$ spaces, modes of convergence, and the inequalities (H\"older, Minkowski, Jensen) that turn the abstract integral into an analytic tool.
\end{remark}
