mistergoneo7

2022-07-05

Let ${A}_{1},{A}_{2},\dots$ be a collection of events, and let $\mathcal{A}$ be the smallest $\sigma$-field of subsets of $\mathrm{\Omega }$ which contains all of them. If $A\in \mathcal{A}$, then there exists a sequence of events $\left\{{C}_{n}\right\}$ such that

where ${\mathcal{A}}_{n}$ is the smallest $\sigma$-field which contains the finite collection ${A}_{1},{A}_{2},\dots ,{A}_{n}.$
However, I do not know how to choose the ${C}_{n}$ such that they turn up in $\sigma \left({A}_{1},..{A}_{n}\right)$?

Elias Flores

Expert

Union of ${\mathcal{A}}_{n}$'s is an algebra which generates $\mathcal{A}$. Just apply Halmos' Theorem now. You get ${C}_{n}\in {\mathcal{A}}_{{k}_{n}}$ for some ${k}_{n}$ incerasing to $\mathrm{\infty }$. But you can keep repeating each ${C}_{n}$ to make sure that ${C}_{n}\in {\mathcal{A}}_{n}$ for each $n$: Look ar $\mathrm{\varnothing },\mathrm{\varnothing },\mathrm{\varnothing },...,{C}_{1},{C}_{1},...{C}_{1},{C}_{2},{C}_{2},...,{C}_{2},...$ where $\mathrm{\varnothing }$ is repeated ${k}_{1}-1$ times, ${C}_{1}$ is repeated ${k}_{2}-{k}_{1}-1$ times, and so on.