10 — MCT for Series, Fatou Applications, Change of Variables, Uniform Integrability

Lecture 10 covers four major topics:

An application of MCT to infinite series of non-negative measurable functions
A tricky Fatou-based argument showing $𝐿^𝑝$ convergence from pointwise convergence + norm convergence,
The change-of-variable formula for pushforward measures,
Uniform integrability, its definition, and its key convergence theorem

1. Example: MCT Applied to Infinite Series

Suppose $g_k \ge 0$ are measurable.
Define the partial sums: $S_n(x) = \sum_{k=1}^n g_k(x).$

Clearly (page 1 of PDF): $S_{n+1}(x) = S_n(x) + g_{n+1}(x) \ge S_n(x).$

Thus $S_n \uparrow S := \sum_{k=1}^\infty g_k$.

By the Monotone Convergence Theorem: $\int S_n \, d\mu \;\uparrow\; \int S \, d\mu.$

Since $\int S_n = \sum_{k=1}^n \int g_k$, we obtain: $\boxed{ \int \Big( \sum_{k=1}^\infty g_k \Big)\, d\mu = \sum_{k=1}^\infty \int g_k\, d\mu. }$

This gives term-by-term integration for non-negative series.

2. A Fatou-Type Trick for Proving $L^p$ Convergence

We are given (page 1):

$p \ge 1$,
$f_n \to f$ almost everywhere,
$\vert f_n\vert _p \to \vert f\vert _p < \infty$.

Goal: $\vert \,f_n - f\,\vert _p \to 0.$

This is not trivial: convergence in $L^p$ does not generally follow from a.e. convergence and convergence of norms, unless convexity is exploited.

Step 1: A convexity inequality

Page 1 contains the inequality: $\left( \frac{\vert x\vert + \vert y\vert }{2} \right)^p \le \frac{\vert x\vert ^p + \vert y\vert ^p}{2}, \qquad p \ge 1.$

Algebraic rearrangement gives: $2^{p-1}\,( \vert x\vert ^p + \vert y\vert ^p ) - \vert x - y\vert ^p \ge 0, \qquad x,y \in \mathbb{R}.$

Apply this to $x = f$, $y = f_n$: $2^{p-1}(\vert f\vert ^p + \vert f_n\vert ^p) - \vert f - f_n\vert ^p \ge 0.$

Thus for all $n$: $\vert f - f_n\vert ^p \le 2^{p-1}(\vert f\vert ^p + \vert f_n\vert ^p).$

Step 2: Apply Fatou to the difference

Rearrange to match Fatou’s lemma (page 1): $2^{p}\,\vert f\vert ^p \le \liminf_{n\to\infty} \left[ 2^{p-1}(\vert f\vert ^p + \vert f_n\vert ^p) - \vert f - f_n\vert ^p \right].$

Integrate and use Fatou:

\[2^{p}\int \vert f\vert^p \le \liminf_n \left[ 2^{p-1} \big( \int\vert f\vert^p + \int\vert f_n\vert^p \big) - \int \vert f - f_n\vert^p \right].\]

But $\vert f_n\vert _p^p = \int\vert f_n\vert ^p \to \vert f\vert _p^p$. Plugging this in yields the right-hand side approaching: $2^{p} \int \vert f\vert ^p - \limsup_n \int \vert f - f_n\vert ^p.$

Hence: $2^{p}\int\vert f\vert ^p \le 2^{p}\int\vert f\vert ^p - \limsup_n \int\vert f - f_n\vert ^p.$

This forces: $\limsup_n \int\vert f - f_n\vert ^p = 0,$ so $\boxed{ \vert f_n - f\vert _p^p = \int\vert f_n - f\vert ^p \to 0. }$

Thus convergence of norms + pointwise a.e. convergence implies full $L^p$ convergence.

3. Change of Variable Formula via Pushforward Measures

Your notes (page 2) emphasize that there is no general change-of-variables formula without defining the correct measure on the target space.

Setup:

$(\Omega_1,\mathcal{F}_1,\mu_1)$,
$(\Omega_2,\mathcal{F}_2)$,
$\varphi: \Omega_1 \to \Omega_2$ measurable.

We want a formula such that: $\int_{\Omega_1} f(\varphi(\omega))\, d\mu_1(\omega) = \int_{\Omega_2} f(y)\, d\mu_2(y).$

But what is $\mu_2$?
From page 2:

Definition (Pushforward measure)

$\boxed{ \mu_2(B) = \mu_1(\varphi^{-1}(B)), \quad B \in \mathcal{F}_2. }$

Then for all measurable $f \ge 0$: $\int_{\Omega_1} f\circ\varphi \ d\mu_1 = \int_{\Omega_2} f\ d\mu_2.$

This is the valid general change-of-variables formula.

Example (from the handwritten notes)

Let:

$\Omega_1 = \Omega$ (probability space),
$\Omega_2 = \mathbb{R}$,
$\varphi = X$ a random variable.

Then the pushforward is: $\mu_2 = P_X, \qquad P_X((a,b]) = P(a < X \le b).$

If $F_X$ is the CDF,
$P_X((a,b]) = F_X(b) - F_X(a).$

Thus: $\mathbb{E}[f(X)] = \int_{\mathbb{R}} f(x)\, dP_X(x),$ and if $P_X$ is absolutely continuous with density $f_X$, this becomes: $\mathbb{E}[f(X)] = \int_{\mathbb{R}} f(x) f_X(x) \, dx.$

4. Uniform Integrability (UI)

Your notes (page 3) introduce the Definition:

Let $(\Omega,\mathcal{F},P)$ be a probability space.
A family ${X_n}$ is uniformly integrable (UI) if: $\phi(M) := \sup_{n\ge 1} \mathbb{E}\left( \vert X_n\vert \, \mathbf{1}_{\{\vert X_n\vert \ge M\}} \right) \longrightarrow 0 \quad\text{as } M\to\infty.$

Intuition (illustrated by the shaded diagram on page 3):
UI controls the tails uniformly in $n$.
The area above $M$ is forced to vanish uniformly.

Key Result

If:

$X_n \to X$ almost surely,
${X_n}$ is uniformly integrable,

then: $\boxed{ \mathbb{E}\vert X_n - X\vert \to 0. }$

Equivalently: $\mathbb{E}[X_n] \to \mathbb{E}[X].$

Sufficient Condition for UI

If there exists an integrable random variable $Y$ such that: $\vert X_n\vert \le Y \quad\text{for all } n,$ then ${X_n}$ is UI.

This is the classical dominating integrable envelope criterion.

5. Connecting DCT and UI

The notes remark (bottom of page 2 and page 3):

DCT requires a single dominating integrable function $g$.
UI generalizes DCT to sequences where domination need not hold pointwise.
UI + a.e. convergence implies convergence in $L^1$, hence convergence of expectations.

Thus:

Tool	Condition	Guarantees
DCT	$\vert X_n\vert\le g$ and $X_n\to X$ a.e., $g\in L^1$	$\mathbb{E}X_n\to\mathbb{E}X$
UI	tail control only	$\mathbb{E}X_n\to\mathbb{E}X$

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