Exercises:Measure Theory - 2016 - 1/Section B/Problem 1

Section B

Problem B1

Part i)

Suppose that [ilmath]\mathcal{A}_n[/ilmath] are algebras of sets satisfying [ilmath]\mathcal{A}_n\subset \mathcal{A}_{n+1} [/ilmath]. Show that [ilmath]\bigcup_{n\in\mathbb{N} }A_n[/ilmath] is an algebra.

• Caution:Is [ilmath]\subset[/ilmath] or [ilmath]\subseteq[/ilmath] desired?

Part ii)

Check that if the [ilmath]A_n[/ilmath] are all sigma-algebras that their union need not be an algebra.

Is a countable union of sigma-algebras (whether monotonic or not) an algebra?

Hint: Try considering the set of all positive integers, [ilmath]\mathbb{Z}_{\ge 1} [/ilmath] with its sigma-algebras [ilmath]\mathcal{A}_n:=\sigma(\mathcal{C}_n)[/ilmath] where [ilmath]\mathcal{C}_n:=\mathcal{P}(\{1,2,\ldots,n\})[/ilmath] where [ilmath]\{1,2,\ldots,n\}\subset\mathbb{N}[/ilmath] and [ilmath]\mathcal{P} [/ilmath] denotes the power set

Check that if [ilmath]\mathcal{B}_1[/ilmath] and [ilmath]\mathcal{B}_2[/ilmath] are sigma-algebras that their union need not be an algebra of sets