Monotone convergence theorem

Remarks

Proof idea: the integrals $\int f_n$ increase and are at most $\int f$, so their limit $L$ satisfies $L \le \int f$. For the reverse bound, fix a simple $s \le f$ and $c \in (0, 1)$. The sets $E_n = \set{f_n \ge c s}$ increase to $X$, so $\int f_n \ge c \int_{E_n} s \to c \int s$ by continuity of measure from below. Letting $c \to 1$ and taking the supremum over $s$ gives $L \ge \int f$.

Example: with $f_n = \min(f, n)$ the theorem computes $\int f$ from bounded functions. For a series of non-negative measurable functions it gives $\int \sum_k g_k \, d\mu = \sum_k \int g_k \, d\mu$.

Both hypotheses matter. Without monotonicity, $f_n = \mathbf{1}_{[n, n+1]}$ on $\R$ tends to $0$ pointwise while each integral is $1$. Without non-negativity, $f_n = -\mathbf{1}_{[n, \infty)}$ increases to $0$ while each integral is $-\infty$. It holds also when $f_n \le f_{n+1}$ only almost everywhere. The analogous fact for sequences of numbers is the monotone convergence theorem for sequences.

Going deeper

For a student

This is the first of the three convergence theorems of Lebesgue integration. Fatou's lemma follows from it, and the dominated convergence theorem follows from Fatou's lemma. It is also the tool to prove that the integral is additive, by approximating with increasing simple functions. In probability it gives $\mathbb{E}[\sum_k Y_k] = \sum_k \mathbb{E}[Y_k]$ for non-negative random variables.