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Let $S$ be a semi-field of subsets of $\Omega$ and $E$ be any subset of $\Omega$. Then, show that $S\cap E$ is a semi-field of subsets of $E$.

I simply need to prove two properties for the collection of subsets of $E$, $S\cap E$:

  1. closed under finite intersection,
  2. for any set $A\in S\cap E$, $A^c$ is a finite disjoint union of sets from $S\cap E$.

The first property is easy to prove as $S$ is a semi-field. For the second part, I tried to proceed, as follows.

Let $A\in S\cap E$. Then, $\exists B\in S$, such that $A=B\cap E$. Writing $A^c=B^c\cup E^c$ and $B=\cup_{i=1}^mC_i$ for some finite $m$ and $C_i\in S~\forall i$ (since $S$ is a semi-field), I wanted to show that $A^c$ can be written as a finite union of sets from $S\cap E$. However, this is not leading to anything in particular. Any help?

zaira
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    You must be concerned with $A^c:=E-S$, i.e. the complement of $A$ wrt $E$ (not wrt $\Omega$). – drhab Feb 18 '23 at 11:42
  • @drhab OH right that makes sense! completely overlooked it. but what do i start with now? – zaira Feb 18 '23 at 11:44

1 Answers1

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Let $\mathcal S$ be a semi-field of subsets of $\Omega$ and let $E\subseteq\Omega$.

Now let $\mathcal S_E:=\{S\cap E\mid S\in\mathcal S\}$.

It must be shown that $\mathcal S_E$ is a semi-field on $E$ and (as you said) being closed under binary intersection is straightforward.

Let $A\in\mathcal S_E$. Then some $S\in\mathcal S$ exists such that $A=S\cap E$. The complement of $A$ wrt to the set $E$ is $S^c\cap E$. Then disjoint sets $S_1,\dots,S_n\in\mathcal S$ exist with $S^c=\bigcup_{i=1}^nS_i$ and a direct consequence of this is that:$$A^c=\bigcup_{i=1}^n(S_i\cap E)$$The sets $S_i\cap E$ are disjoint elements of $\mathcal S_E$ so we are ready.

drhab
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