How do we prove the union bound inequality (Boole's inequality) for infinite sets?

Question

I am aware that Boole's inequality can be proved by induction. How do we extend these results to an infinite set? (since induction cannot be applied in this case)

P($\bigcup\limits_{i=1}^{\infty} A_i$) $\leq$ $\sum \limits_{i=1}^{\infty} P(A_i)$

Answer 1

Define $B_i = A_i \backslash \bigcup^{i-1}_{j=1} A_j$, then we see that $\bigcup^{i}_{j=1} B_j=\bigcup^{i}_{j=1} A_j$, so $\bigcup^{\infty}_{j=1} B_j=\bigcup^{\infty}_{j=1} A_j$. Also note that $B_j\subseteq A_j$, so $ \mathbb P (B_j) \leq \mathbb P(A_j)$ and thus $$\mathbb P (\bigcup^{\infty}_{j=1}A_j) = \mathbb P (\bigcup^{\infty}_{j=1}B_j) = \sum^{\infty}_{j=1} \mathbb P (B_j) \leq \sum^{\infty}_{j=1} \mathbb P(A_j)$$ where the second equality holds due to all $B_j$ being pairwise disjoint.

Answer 2

Alternatively, let $f(X)$ be the indicator that $X\in\bigcup_i A_i$ and let $g(X)$ denote the number of sets $A_i$ that contain $X$; it might for some $X$ values be infinite. Clearly $f(X)\le g(X)$ with probability $1$. Take expectations, so $E(f(X))\le E(g(X))$.