Exercise 0.11 (Measurable induction)

Let $\mathcal{𝒜}$ be a collection of subsets of $\mathrm{\Omega }$, and let $p(E)$ be a property of subsets $E$ of $\mathrm{\Omega }$ (thus $P(E)$ is true or false for each $E\in \mathrm{\Omega }$). Assume the following axioms:

1.

$P(\varnothing )$ is true.

2.

If $E\subseteq \mathrm{\Omega }$ is such that $P(E)$ is true, then $P(\mathrm{\Omega }∕E)$ is also true.

3.

If $E_1,E_2,\dots \subseteq \mathrm{\Omega }$ are such that $P(E_n)$ is true for all $n\in \mathbb{N}$, then $P\left({\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_{n\in \mathbb{N}}{E}_n\right)$ is true.

Show that if $P(E)$ is true for all $E\in \mathcal{𝒜}$ then $P(E)$ is true for all $E\in ⟨\mathcal{𝒜}⟩$.