Let $\mathcal{F}$ be a $\sigma$-algebra corresponding to a sample space $\Omega$. Let $H$ be a subset of $\Omega$ that does not belong to $\mathcal{F}$. Consider the collection $\mathcal{G}$ of all sets of the form $(H\cap A)\cup (H^C \cap B)$, where $A,B\in \mathcal{F}$. Show that $\mathcal{G}$ is a $\sigma$-algebra.

I noticed that $\mathcal{F}\subseteq \mathcal{G}$ and also that $\Omega\in \mathcal{G}$ but I am having difficulty in proving that

  1. $A\in \mathcal{G}\implies A^C\in\mathcal{G}$

  2. $A_1,A_2,A_3,\cdots \in\mathcal{G}\implies \bigcup_{n=1}^\infty A_n\in \mathcal{G}$

  1. follows from $$((H \cap A) \cup (H^C \cap B))^C = (H \cap A^C) \cup (H^C \cap B^C)$$ (this identity can be proved using Venn diagrams)

  2. follows from $$\bigcup_n((H \cap A_n) \cup (H^C \cap B_n)) = \left( H \cap \bigcup_n A_n \right) \cup \left( H^C \cap \bigcup_n B_n \right)$$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.