Lebesgue Integration - Main Theorem

TheoremMonotone Convergence Theorem (MCT)

Let $(X, \mathcal{F}, \mu)$ be a measure space. Suppose $\{f_n\}$ is a sequence of non-negative measurable functions such that: $0 \leq f_1(x) \leq f_2(x) \leq f_3(x) \leq \cdots$ for all $x \in X$ . Let $f(x) = \lim_{n \to \infty} f_n(x)$ be the pointwise limit. Then: $\lim_{n \to \infty} \int_X f_n \, d\mu = \int_X f \, d\mu$

In other words, for monotone increasing sequences of non-negative functions, the limit and integral can be interchanged.

The Monotone Convergence Theorem is arguably the most important result in integration theory. It allows us to pass limits inside integrals under very mild conditions, without requiring uniform convergence or even domination.

ExampleApplication to Series

Consider the sum $\sum_{n=1}^{\infty} a_n$ of non-negative terms. Using counting measure on $\mathbb{N}$ , let $f_N = \sum_{n=1}^{N} a_n \chi_{\{n\}}$ . Then $f_N \uparrow f$ where $f(n) = a_n$ .

By MCT: $\sum_{n=1}^{\infty} a_n = \int_\mathbb{N} f \, d\mu = \lim_{N \to \infty} \int_\mathbb{N} f_N \, d\mu = \lim_{N \to \infty} \sum_{n=1}^{N} a_n$

This shows that MCT generalizes the theory of infinite series.

ExampleComputing Integrals via Approximation

To compute $\int_0^\infty e^{-x} \, d\lambda$ , define $f_n = e^{-x} \chi_{[0,n]}$ . Then $f_n \uparrow e^{-x}$ , so: $\int_0^\infty e^{-x} \, d\lambda = \lim_{n \to \infty} \int_0^n e^{-x} \, dx = \lim_{n \to \infty} (1 - e^{-n}) = 1$

Remark

Important consequences of MCT include:

Beppo Levi's Theorem: For non-negative measurable functions $g_n$ : $\int_X \sum_{n=1}^{\infty} g_n \, d\mu = \sum_{n=1}^{\infty} \int_X g_n \, d\mu$

This follows by applying MCT to the partial sums $f_N = \sum_{n=1}^{N} g_n$ .

Fatou's Lemma: For non-negative measurable $f_n$ : $\int_X \liminf_{n \to \infty} f_n \, d\mu \leq \liminf_{n \to \infty} \int_X f_n \, d\mu$

This is derived by applying MCT to $g_k = \inf_{n \geq k} f_n \uparrow \liminf f_n$ .

The proof of MCT uses the definition of the integral as a supremum over simple functions, combined with the monotonicity property to show convergence from both directions.