Fatou’S Theorem

1. Core Concept Overview & Intuition

Fatou’s Theorem is a foundational convergence result in measure-theoretic probability. It addresses a central question: if a sequence of random variables converges in some sense, when may we interchange the operations of taking a limit and taking an expectation? The theorem provides one-sided inequalities that bound the expectation of a limit inferior (or limit superior) by the limit inferior (or limit superior) of the expectations. These inequalities hold under a suitable integrability condition that prevents mass from “escaping to infinity.”

The intuition is best grasped through the notion of a monotone envelope. Given a sequence ${X_n}$, define the running infimum $Y_n=\inf_{k\ge n}X_k$. The sequence ${Y_n}$ is non-decreasing and converges to $\liminf_n X_n$. Because each $Y_n$ is bounded above by $X_n$, the expectation of $Y_n$ cannot exceed the expectation of $X_n$. Passing to the limit and invoking the monotone convergence theorem yields the first inequality. The second inequality is obtained by applying the first to the reflected sequence $Z-X_n$, where $Z$ is an integrable dominating upper bound.

The theorem is deliberately stated as an inequality rather than an equality. Equality can fail when probability mass drifts toward regions where the random variables become large but increasingly rare. The integrability of a lower bound $Y$ (or upper bound $Z$) rules out such pathological behaviour by anchoring the sequence from below (or above) with an integrable random variable.

The subtopics listed in the syllabus — definitions, sequences and limits, fields, DeMorgan’s rule, functions, probability, approaches to probability, and expectation — supply the language and machinery on which Fatou’s Theorem rests. Sets and their sections, measurable spaces, the algebra of events, the behaviour of limits, and the definition of expectation as an integral are all prerequisites. The theorem itself sits at the intersection of these ideas: it is a statement about the expectation functional, which is defined on a field of events and extended to random variables via integration.

2. Mathematical Formalization & Core Definitions

> Definition (Random Variable and Expectation). A random variable $X$ is a measurable function from a probability space to $\mathbb{R}$. Its expectation $E(X)$ is defined as the integral of $X$ with respect to the probability measure, whenever the integral exists. For a non-negative random variable, $E(X)$ always exists (possibly infinite). A random variable is integrable if $E\lvert X\rvert<\infty$.

> Definition (Limit Inferior and Limit Superior of a Sequence). For a sequence of real numbers ${x_n}$, the limit inferior is

$$
\liminf_{n}x_n=\sup_{n\ge 1}\inf_{k\ge n}x_k,
$$

> and the limit superior is

$$
\limsup_{n}x_n=\inf_{n\ge 1}\sup_{k\ge n}x_k.
$$

> The limit inferior is the smallest accumulation point, and the limit superior is the largest. When they coincide, the sequence converges to their common value.

> Definition (Almost Sure Convergence). A sequence ${X_n}$ converges almost surely to $X$ if the set of outcomes for which $X_n$ does not converge to $X$ has probability zero. In symbols, $P(\lim_n X_n=X)=1$.

> Definition (Convergence in Probability). A sequence ${X_n}$ converges in probability to $X$ if for every $\epsilon>0$,

$$
P(\lvert X_n-X\rvert>\epsilon)\to 0\quad\text{as }n\to\infty.
$$

> Definition (Monotone Convergence Theorem). Let ${X_n}$ be a sequence of non-negative random variables that increases monotonically to $X$, written $X_n\uparrow X$. Then

$$
E(X_n)\uparrow E(X).
$$

> Definition (Markov Inequality). For a non-negative random variable $X$ and $a>0$,

$$
P(X\ge a)\le\frac{E(X)}{a}.
$$

> This inequality is used to deduce convergence in probability from convergence of absolute moments.

> Definition (Integrable Lower and Upper Bounds). A random variable $Y$ is an integrable lower bound for a sequence ${X_n}$ if $Y\le X_n$ for all $n$ and $E\lvert Y\rvert<\infty$. Similarly, $Z$ is an integrable upper bound if $X_n\le Z$ for all $n$ and $E\lvert Z\rvert<\infty$.

> Definition (Field of Events). A field (algebra) $\mathcal{F}$ on a sample space $\Omega$ is a collection of subsets of $\Omega$ closed under finite unions, finite intersections, and complementation, and containing $\Omega$. A probability measure $P$ on $\mathcal{F}$ assigns to each event $A\in\mathcal{F}$ a number $P(A)\in[0,1]$ with $P(\Omega)=1$ and countable additivity on disjoint events.

> Definition (DeMorgan’s Rule). For any collection of sets ${A_i}$,

$$
\left(\bigcup_i A_i\right)^c=\bigcap_i A_i^c,\qquad \left(\bigcap_i A_i\right)^c=\bigcup_i A_i^c.
$$

> These identities are used throughout to manipulate events such as $\lvert X_n-X\rvert>\epsilon$ and their complements.

> Definition (Functions and Sections). A function $f$ defined on a product space $U_1\times U_2$ has $w_1$-section $f_{w_1}$ defined on $U_2$ by $f_{w_1}(w_2)=f(w_1,w_2)$, and similarly for $w_2$-sections. These notions underpin the product-space language used in the surrounding theory, though the present topic uses only the one-dimensional expectation framework.

> Definition (Approaches to Probability). Probability may be approached through the classical (equally likely outcomes), frequentist (long-run relative frequency), and axiomatic (Kolmogorov) viewpoints. The axiomatic approach, built on a field of events and a probability measure, is the setting in which Fatou’s Theorem is stated.