Continuity of probability measures

Statement

Let $(\Omega, \mathcal{F}, P)$ be a probability space. If $A_1 \subseteq A_2 \subseteq \cdots$ are events and $A = \bigcup_{n=1}^{\infty} A_n$, then $\lim_{n \to \infty} P(A_n) = P(A)$. If $B_1 \supseteq B_2 \supseteq \cdots$ are events and $B = \bigcap_{n=1}^{\infty} B_n$, then $\lim_{n \to \infty} P(B_n) = P(B)$.

Remarks

The first part is continuity from below. Its proof writes $A$ as the disjoint union of $A_1$ and the sets $A_{n+1} \setminus A_n$, and uses countable additivity. The second part, continuity from above, follows from the first part with the complements $B_n^c$. For a general measure $\mu$, continuity from above needs $\mu(B_1) < \infty$. A probability measure is finite, so this holds. For example, $P(X \le x + 1/n) \to P(X \le x)$ for a real random variable $X$. This gives the right continuity of the cumulative distribution function.

Explanations

L1 human

If events grow step by step, their probabilities grow toward the probability of the limit event. For example, the chance that a fair coin shows heads at least once in the first $n$ tosses is $1 - 2^{-n}$. It tends to the chance $1$ of heads at some toss.