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

Section B

Problem B1

Part i)

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

• Caution:Is $\subset$ or $\subseteq$ desired?

Part ii)

Check that if the $A_n$ 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, $\mathbb{Z}_{\ge 1}$ with its sigma-algebras $\mathcal{A}_n:=\sigma(\mathcal{C}_n)$ where $\mathcal{C}_n:=\mathcal{P}(\{1,2,\ldots,n\})$ where $\{1,2,\ldots,n\}\subset\mathbb{N}$ and $\mathcal{P}$ denotes the power set

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