the dominated convergence theorem

This is the workhorse theorem for swapping a limit with an expectation, and the one you reach for most often in practice. It does not require the variables to increase (unlike monotone convergence) or to be non-negative (unlike the basic case). Instead it asks for a single, simple safeguard: that the whole sequence stays trapped underneath one fixed, integrable ceiling. If you can find such a ceiling, the swap is guaranteed.

Precisely: suppose X1, X2, X3, ... converges pointwise to X, and there is a single dominating variable Y with E[Y] finite such that |Xn| <= Y for every n and every outcome. Then E[Xn] converges to E[X], so lim of E[Xn] = E[lim of Xn]. The role of Y is to prevent the mischief that wrecks the swap in general — namely probability mass escaping to infinity. Because none of the Xn can stick out above Y, no mass can run off to where the integral cannot see it, and so the limit of the integrals must equal the integral of the limit. The proof is a clean two-line application of Fatou's lemma to Y + Xn and Y - Xn, which pinches the limit from both sides.

In probability this is the everyday tool: it justifies differentiating under the expectation sign, taking limits inside E[...], and proving E[Xn] -> E[X] for approximating sequences — for instance showing that characteristic functions, moment generating functions, and densities behave continuously. The single honest warning is that the dominating Y must be integrable and must not depend on n; a bound that grows with n, or a finite bound at each n with no common integrable ceiling, is not enough, and then the conclusion can genuinely fail (the escaping-spike example shows exactly this).