monotone convergence theorem

If you fill a bathtub by adding more and more water and never draining any, the final water level is just the limit of the rising levels — no surprises. The monotone convergence theorem says the same for integrals: if non-negative functions only ever increase toward a limit, the integral of the limit is the limit of the integrals. Increasing and non-negative are the price; in exchange the limit slides right through the integral sign.

Precisely: let f_1 less than or equal to f_2 less than or equal to ... be measurable functions with each f_n greater than or equal to 0, converging pointwise (almost everywhere) to f. Then the integral of f equals the limit as n goes to infinity of the integral of f_n. Both sides may be plus infinity, and the theorem still holds; no domination or finiteness is assumed.

Two warnings on the hypotheses, both genuine. Without non-negativity it can fail; and without monotone increase it can fail dramatically — the moving-bump example, f_n = the indicator of [n, n+1], has every integral equal to 1 but converges pointwise to 0, whose integral is 0. The theorem is the workhorse behind Fatou's lemma and the definition of the integral itself.