ThmDex – An index of mathematical definitions, results, and conjectures.
Proof P3673 on R5393: I.I.D. real empirical distribution measure converges to a probability for a fixed Borel set
P3673
Since $\mathbb{E} I_{X_1 \in B} = \mathbb{P}(X_1 \in B) < \infty$, this result is a special case of R5399: Sample average of I.I.D. integrable random real numbers converges to expectation almost surely. $\square$