beatricalwu

2022-07-21

Let $\mathcal{A}$=event space. Assume that $\left\{{A}_{n}\right\}$ is a monotone nondecreasing sequence of events in A$\mathcal{A}$. Let $\left\{{A}_{n}\right\}$ be a sequence of sets defined as follows:

Prove that $\left\{{B}_{n}\right\}$ is a sequence of mutually exclusive events in $\mathcal{A}$. Justify each line of proof.
I know that what I need to show here is ${B}_{i}\cap {B}_{j}=\mathrm{\varnothing }$ for any $i. I've divided it to be proof by cases where case $1$ is and case $2$ is $i\in \left\{2,3,4,\dots \right\}$ and $j\in \left\{2,3,4,\dots \right\}$ where $i so that and where $i\in \left\{2,3,4\dots \right\}$ and $j\in \left\{i+1,i+2,\dots \right\}$.

tykoyz

Your idea is correct. As a first step, take an element $e\in {B}_{j}$. You have to prove that $e\notin {B}_{i}$. For this, use the definition ${B}_{j}={A}_{j}-{A}_{j-1}$. Since $e\in {A}_{j}-{A}_{j-1}$, what does this tell you about $e$?

Kenya Leonard

If $i then ${B}_{i}\subseteq {A}_{i}\subseteq {A}_{j-1}$ and ${A}_{j-1}\cap {B}_{j}=\varnothing$.