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

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  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 $$\bigcup_n((H \cap A_n) \cup (H^C \cap B_n)) = \left( H \cap \bigcup_n A_n \right) \cup \left( H^C \cap \bigcup_n B_n \right)$$

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