14 — Fubini’s Theorem and Applications
This lecture is essentially a full constructive proof of Fubini’s Theorem, including:
Construction of the product measure,
The λ–system / π–system argument for measurability of sections,
Proof of Tonelli (non-negative case) and then Fubini (integrable case),
Applications:
Expectation identity
𝐸 [ 𝑋 𝑝 ] = ∫ 0 ∞ 𝑝 𝑥 𝑝 − 1 𝑃 ( 𝑋 ≥ 𝑥 ) 𝑑 𝑥 E[X p ]=∫ 0 ∞
px p−1 P(X≥x)dx,
A more advanced example involving two distribution functions 𝐹 F and 𝐺 G, showing how integration by parts in probability is just Fubini in disguise.
We consider two σ-finite measure spaces:
\[(\Omega_1,\mathcal{F}_1,\mu_1), \qquad (\Omega_2,\mathcal{F}_2,\mu_2),\]and the product space:
\[\Omega = \Omega_1 \times \Omega_2, \qquad \mathcal{F} = \mathcal{F}_1 \otimes \mathcal{F}_2, \qquad \mu = \mu_1 \times \mu_2.\]A function on the product is written $f(x,y)$, with
$x\in\Omega_1,\,y\in\Omega_2$.
1. Statement of Fubini’s Theorem
If $f:\Omega\to\mathbb{R}$ is measurable and either
- $f\ge 0$, or
- $\displaystyle \int_\Omega \vert f\vert \,d\mu<\infty$,
then:
\[\int_{\Omega_1} \left( \int_{\Omega_2} f(x,y)\, d\mu_2(y) \right) d\mu_1(x) = \int_{\Omega} f\, d\mu = \int_{\Omega_2} \left( \int_{\Omega_1} f(x,y)\, d\mu_1(x) \right) d\mu_2(y).\]Tonelli = “$f\ge 0$” case.
Fubini = “$\int \vert f\vert <\infty$” case.
2. Construction of Product Measure (Review)
We start with the algebra:
\[\mathcal{L} = \{ \bigcup_{i=1}^m A_i \times B_i : A_i\in\mathcal{F}_1,\, B_i\in\mathcal{F}_2,\, (A_i\times B_i)\text{ disjoint} \}.\]Define
\[M(A\times B)=\mu_1(A)\mu_2(B).\]By Carathéodory, $M$ extends uniquely to a measure $\mu$ on
$\mathcal{F}=\mathcal{F}_1\otimes\mathcal{F}_2$.
If $\mu_1,\mu_2$ are σ-finite, so is $\mu$ (noted on page 1 of the handwritten notes).
3. Proof of Fubini — Step Structure
The notes outline the standard four-step method.
Step 1. Indicator of a measurable set
Let $f = \mathbf{1}_D$ with $D \in \mathcal{F}$.
Show:
Step 2. Non-negative simple functions
Let
\(h=\sum_{i=1}^n c_i\, \mathbf{1}_{D_i},
\qquad c_i\ge 0,
\quad D_i\in\mathcal{F}.\)
By linearity and Step 1:
\[\int h = \sum_i c_i \mu(D_i) = \int_{\Omega_1}\!\int_{\Omega_2} h\, d\mu_2 d\mu_1.\]Step 3. Non-negative measurable $f$
Let $0\le h_n\uparrow f$ be simple (the dyadic construction is drawn on page 1):
\[h_n(x,y) = \frac{k}{2^n}\quad \text{if }\frac{k}{2^n} \le f < \frac{k+1}{2^n}.\]Then by MCT:
\[\int f = \lim_n \int h_n = \lim_n \int_{\Omega_1}\!\int_{\Omega_2} h_n = \int_{\Omega_1}\!\int_{\Omega_2} f.\]This proves Tonelli.
Step 4. Integrable $f$
Write:
\[f = f^+ - f^-, \qquad f^+,f^-\ge 0.\]Both are integrable since $\int\vert f\vert <\infty$.
Apply Tonelli to each:
Thus Fubini holds.
4. Sections and Measurability (Lemmas 1 and 2)
On page 2, the notes move “back to Step One” to justify measurability of the section maps $E_x$.
Given $E\in\mathcal{F}$,
- The section at $x$ is
\(E_x = \{y\in\Omega_2 : (x,y)\in E\}.\) - For rectangles $E=A\times B$:
\(E_x= \begin{cases} B,& x\in A \\ \varnothing,& x\notin A. \end{cases}\)
Lemma 1.
If $E \in \mathcal{F}$, then for each fixed $x$, $E_x \in \mathcal{F}_2$.
Similarly for $E^y\in\mathcal{F}_1$.
Lemma 2.
If $E\in\mathcal{F}$, then \(g(x) = \mu_2(E_x)\) is $\mathcal{F}_1$-measurable, and \(\int_{\Omega_1} g(x)\, d\mu_1(x) = \mu(E).\)
Proof idea (page 2):
Use the π–λ theorem:
- Show the class of all $E$ for which Lemma 2 holds is a λ–system.
- Rectangles form a π–system contained in it.
- By Dynkin’s π–λ theorem, all of $\mathcal{F}$ satisfies the lemma.
5. Application of Fubini: Expectation Formula
Let $X\ge 0$ be a random variable on $(\Omega,\mathcal{F},P)$, let $p>0$.
The notes (page 2–3) show:
Derivation (page 3):
Use: \(X^p = \int_0^{X} p x^{p-1}\, dx = \int_0^\infty p x^{p-1}\, \mathbf{1}_{\{x\le X\}}\, dx.\)
Apply Fubini:
\[\mathbb{E}[X^p] = \int_\Omega \int_0^\infty p x^{p-1}\mathbf{1}_{\{x\le X(\omega)\}}\, dx\, dP(\omega) = \int_0^\infty p x^{p-1} P(X\ge x)\, dx.\]The picture on page 3 (red arrows) shows switching the order of integration in the region
${(x,\omega): 0\le x \le X(\omega)}$.
6. A More Advanced Example:
Double Stieltjes Integration and Jump Terms
On page 3 the notes consider:
- $F(x)=\mu_1((-\infty,x])$,
- $G(y)=\mu_2((-\infty,y])$.
Drawn is a rectangle $[a,b]\times[a,b]$ split into diagonal $A$ and the off-diagonal regions $B$.
The key identities:
\[\mu(A) = \int_a^b (F(y)-F(a))\, dG(y),\] \[\mu(B) = \int_a^b (G(b)-G(y))\, dF(y).\]Adding them (page 3 boxed identity):
\[\mu(A)+\mu(B) = [F(b)G(b)-F(a)G(a)] + \mu(\{(x,x): a<x\le b\}).\]The diagonal term involves the jump sizes:
\[\mu(\{(x,x)\}) = \sum_{a< x\le b} \Delta F(x)\cdot \Delta G(x).\]Thus the notes highlight:
- “finite # of jumps will survive,”
- Stieltjes integrals capture both continuous parts and jump contributions.
This is the measure-theoretic version of “integration by parts” for CDFs.
7. Summary of Lecture 14
- Built product measures and proved Fubini via:
indicator → simple → positive → general integrable. - Showed section measurability via π–λ arguments.
- Applied Fubini to derive important expectation formulas.
- Illustrated how multivariate Stieltjes integration includes jump terms, explaining why Fubini is “better than integration by parts.”
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