dominated convergence theorem

This is the theorem you reach for whenever you want to swap a limit and an integral. The danger is mass escaping to infinity (recall Fatou's strict inequality). The cure: trap the whole sequence under one fixed integrable umbrella. If a single integrable function dominates all of them in absolute value, no mass can leak away, and the limit passes cleanly inside the integral sign.

Statement: let f_n be measurable functions converging pointwise almost everywhere to f, and suppose there is an integrable g with |f_n| less than or equal to g for all n. Then f is integrable, and the integral of f equals the limit of the integrals of the f_n. Moreover the integral of |f_n - f| tends to 0, i.e. convergence holds in L^1, which is even stronger than convergence of the integrals.

The dominating function g is the crux; without it the conclusion can fail, as the tall-spike and moving-bump examples show. A common refinement: domination by a sequence g_n converging in L^1 also suffices (generalized DCT). This theorem is why differentiating under the integral sign and interchanging sum and integral are usually legal — find a g, and you are done.