Alannah Short

2022-06-26

Let $\left(X,\mathcal{A}\right)$ y $\left(Y,\mathcal{B}\right)$ measure spaces. Let $R:=R\left(\mathcal{A},\mathcal{B}\right)$ the collection of measurable rectangles $R:=\left\{A×B:A\in \mathcal{A},B\in \mathcal{B}\right\}.$. Prove that the algebra of subsets of $X×Y$ generated by $R$ is the collection of finite unions of elements of $R$. I have a hint and it is in which I call $C$ to be the collection of finite unions of elements of $R$, prove that $C$ is in the algebra generated by $R$, then prove that $C$ is an algebra that contains $R$.

### Answer & Explanation

EreneDreaceaw

You only need to prove that the set $\mathcal{A}$ of finite disjoint unions of elements of $R$ is an algebra.
Note that $R$ has the following properties:
$\mathrm{\varnothing }\in R,$
$E,F\in R\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}E\cap F\in R,$

Any set $R$ that satisfies the above three hypotheses has the property that the algebra generated by $R$ is the collection of finite disjoint unions of elements of $R$. This is not difficult to verify.

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