# Limit of the Expectation of Monotonically Decreasing Sequence of Non-Negative R.V.s

The Monotone Convergence Theorem states: Let $$(X_n)_n$$ be a sequence of random variables which are non-negative and increasingly converge to a random variable $$X$$, meaning that $$0 \leq X_1 \leq X_2 \leq ... \leq X$$ with $$\lim_{n \rightarrow \infty} X_n = X$$. Then, we can interchange limit with expectations: $$\lim_{n\rightarrow \infty} EX_n = E[\lim_{n \rightarrow \infty} X_n ] = EX.$$ I would wish to inquire under what condition(s) this 'interchange' could occur if instead given a sequence of random variables $$(X_n)_n$$ which are non-negative and decreasingly converge to $$X$$.

It would be enough for $$EX_1 < \infty$$. Indeed, if $$X_n$$ decrease to $$X$$, then since the sequence of random variables $$X_1 - X_n$$ are positive and increasing to $$X_1 - X$$, the MCT guarantees that $$E(X_1 - X_n) \to E(X_1 - X),$$ and since $$0 \leq EX \leq EX_n \leq EX_1 < \infty$$ one can now use linearity to recover $$EX_n \to EX$$.