Dominated Convergence Theorem (Lebesgue)

1. Core Concept Overview & Intuition

The Dominated Convergence Theorem (Lebesgue) is the central tool that lets us pass a limit through an expectation (or an integral). In words: if a sequence of random variables converges to a limit random variable, and every member of the sequence is trapped beneath a single integrable random variable, then the expectations converge to the expectation of the limit.

The intuition rests on two ingredients working together.

• Convergence supplies the approach of $X_n$ to $X$, so that for each outcome the values $X_n$ eventually sit close to $X$.
• Domination supplies a uniform ceiling $Y$ that is integrable. Because no $X_n$ can escape above $Y$ in absolute value, the mass of the sequence cannot drift off to infinity or concentrate on thin sets. This prevents the failure mode in which expectations converge to something other than the expectation of the limit.

The theorem is best understood as a sufficient condition for the interchange of limit and expectation:

$$
\lim_{n\to\infty} E(X_n) = E(X).
$$

The domination hypothesis $\lvert X_n\rvert \le Y$ with $Y$ integrable is precisely what rules out the escape of probability mass. Without it, convergence alone does not guarantee convergence of expectations.

The surrounding material in this unit builds the supporting machinery. Sequences and limits give the notion of convergence used in the hypothesis. The product-space construction and the mutual-convergence and closure results show that convergence in probability is stable under the arithmetic operations that appear inside expectations. These supporting results are what make the dominated convergence argument usable in practice.

2. Mathematical Formalization & Core Definitions

2.1 Definitions

A random variable $X$ is a measurable function from a probability space into $\mathbb{R}$. Its expectation $E(X)$ is the integral of $X$ with respect to the underlying probability measure. A random variable is integrable when $E(\lvert X\rvert) < \infty$.

A sequence of random variables ${X_n}$ converges in probability to $X$, written $X_n \xrightarrow{p} X$, if for every $\epsilon > 0$,

$$
P\bigl[\,\lvert X_n - X\rvert > \epsilon\,\bigr] \to 0 \quad \text{as } n \to \infty.
$$

A sequence ${X_n}$ converges almost surely to $X$ if $X_n \to X$ with probability one. A consequence of almost-sure convergence is that the limit inferior and limit superior both agree with $X$:

$$
\liminf_{n} X_n = \limsup_{n} X_n = X
$$

holds with probability one.

2.2 Sequences and Limits

The limit inferior and limit superior of a sequence of random variables are defined pointwise:

$$
\liminf_{n} X_n = \sup_{n} \inf_{k \ge n} X_k, \qquad \limsup_{n} X_n = \inf_{n} \sup_{k \ge n} X_k.
$$

Almost-sure convergence is the statement that these two agree with $X$ almost everywhere. The dominated convergence theorem is stated for convergence in probability.

2.3 DeMorgan's Rule

For any collection of sets ${A_i}$,

$$
\Bigl(\bigcup_i A_i\Bigr)^{c} = \bigcap_i A_i^{c}, \qquad \Bigl(\bigcap_i A_i\Bigr)^{c} = \bigcup_i A_i^{c}.
$$

DeMorgan's rule converts statements about "at least one" into statements about "none", and is used repeatedly to rewrite events such as ${\lvert X_n - X\rvert > \epsilon}$ in complementary form.

2.4 Functions and Sections

A function $f: U_1 \times U_2 \to \mathbb{R}$ is measurable when preimages of measurable sets are measurable. For a fixed $w_1 \in U_1$, the $w_1$-section of $f$ is the function $f_{w_1}$ defined by

$$
f_{w_1}(w_2) = f(w_1, w_2),
$$

and similarly the $w_2$-section is $f_{w_2}(w_1) = f(w_1, w_2)$. Sections are the device by which a function of two variables is studied one coordinate at a time.

2.5 Probability and Conditional Probability

The compound event $AB$ (also written $A \cap B$) satisfies

$$
P(AB) = P(A)\,P(B \mid A),
$$

where $P(B \mid A)$ is the conditional probability of $B$ given $A$:

$$
P(B \mid A) = \frac{P(AB)}{P(A)}.
$$

Events $A$ and $B$ are independent when $P(B \mid A) = P(B)$, equivalently $P(AB) = P(A)P(B)$.

2.6 Expectation

For a random variable $X$ with distribution function $F$, the expectation is the integral

$$
E(X) = \int x \, dF(x).
$$