Let $\mathcal{A}$ be a $\sigma$-algebra of subsets of $\Omega$ and let $B\in\mathcal{A}$. Show that $\mathcal{F}=\{A\cap B: A \in\mathcal{A}\}$ is a $\sigma$-algebra of subsets of $B$. Is it still true when $B$ is a subset of $\Omega$ that does not belong to $\mathcal{A}$?

Homepage › Solution manuals › Jean Jacod › Probability Essentials › Exercise 2.6 (Restriction of a $\sigma$-algebra)

Exercise 2.6 (Restriction of a $\sigma$-algebra)

Let $𝒜$ be a $σ$ -algebra of subsets of $Ω$ and let $B \in 𝒜$ . Show that $ℱ = {A \cap B : A \in 𝒜}$ is a $σ$ -algebra of subsets of $B$ . Is it still true when $B$ is a subset of $Ω$ that does not belong to $𝒜$ ?

Exercise 2.6 (Restriction of a $\sigma$-algebra)

Add answer