Convergence Theorems - Main Theorem

TheoremBounded Convergence Theorem

Let $(X, \mathcal{F}, \mu)$ be a finite measure space (i.e., $\mu(X) < \infty$ ). Suppose $\{f_n\}$ is a sequence of measurable functions such that:

$f_n(x) \to f(x)$ for almost every $x \in X$
There exists $M > 0$ such that $|f_n(x)| \leq M$ for all $n$ and almost every $x$

Then $f$ is integrable and: $\lim_{n \to \infty} \int_X f_n \, d\mu = \int_X f \, d\mu$

The Bounded Convergence Theorem is a special case of the Dominated Convergence Theorem where the dominating function is the constant $M$ and the measure space is finite. Its simplicity makes it a useful tool for finite measure spaces.

ExampleApplication to Probability

In probability theory, where $\mu$ is a probability measure (so $\mu(X) = 1 < \infty$ ), the Bounded Convergence Theorem is particularly useful.

Suppose $X_n$ are random variables (measurable functions) with $|X_n| \leq M$ almost surely, and $X_n \to X$ almost surely. Then: $\mathbb{E}[X] = \lim_{n \to \infty} \mathbb{E}[X_n]$

This applies to bounded random variables on probability spaces, such as indicators, truncated variables, or variables taking values in compact sets.

Remark

The Bounded Convergence Theorem can be seen as a corollary of DCT:

The dominating function is $g(x) = M$
Since $\mu(X) < \infty$ , we have $\int_X g \, d\mu = M \cdot \mu(X) < \infty$
Thus $g$ is integrable, and DCT applies

However, BCT is often stated separately because its hypotheses are simpler to verify: just check that the functions are uniformly bounded and the measure space is finite.

Counterexample without finite measure: On $\mathbb{R}$ with Lebesgue measure, let $f_n = \chi_{[n, n+1]}$ . Then $|f_n| \leq 1$ and $f_n \to 0$ pointwise, but $\int f_n = 1 \not\to 0 = \int 0$ . The failure occurs because $\lambda(\mathbb{R}) = \infty$ .

The Bounded Convergence Theorem is particularly valuable in probability and statistics, where probability measures are finite by definition, and many random variables of interest are bounded or can be truncated to bounded versions for approximation purposes.