the monotone convergence theorem
A recurring need in probability is to swap a limit with an integral: you have a sequence of expectations E[X1], E[X2], E[X3], ... and you want to know whether their limit equals the expectation of the limit, E[lim Xn]. In general this swap can fail, and getting it wrong leads to false conclusions. The monotone convergence theorem identifies the simplest situation where the swap is always legal: when the variables only ever increase.
Precisely: if X1, X2, X3, ... is a sequence of non-negative random variables that increases pointwise (X1 <= X2 <= X3 <= ... for every outcome) to a limit X, then E[Xn] increases up to E[X]. In symbols, lim of E[Xn] = E[lim of Xn] = E[X], even allowing the value +infinity. The picture is that as each Xn fills in more and more of the area under X from below, its integral climbs steadily toward the full integral and cannot overshoot or get stuck short. No extra condition is needed beyond non-negativity and the increasing structure — that monotone increase is exactly what blocks the mass from leaking away.
This theorem is the foundation stone of the whole Lebesgue integral: the integral of a non-negative function is defined as a limit of integrals of increasing simple functions, and MCT is what makes that definition consistent and well-behaved. In probability it justifies expanding an expectation as an increasing sum (so E of an infinite sum of non-negative terms equals the sum of their expectations), and it underlies the construction of conditional expectation and many limit arguments. The honest caveat: drop monotonicity or non-negativity and the swap can genuinely fail, which is why Fatou's lemma and dominated convergence exist for the other cases.
Let Xn = min(X, n) for a non-negative X. As n grows, X1 <= X2 <= ... increases up to X, so MCT gives E[X] = lim E[min(X, n)]. This is the standard way to define E[X] for an unbounded non-negative variable as the limit of expectations of truncated versions.
When variables increase to a limit, their expectations increase to the limit's expectation — no leakage possible.
Both conditions matter: the sequence must be non-negative and increasing. Without them mass can escape to infinity and the limit of expectations can be strictly less than the expectation of the limit.