I'm trying to Solve the following question:

Let $X$ be a non empty set and $\mathcal R$ be a $\sigma$-ring from subset of $X$. Prove that: $$S=\{E\subset X; E\cap F \in \mathcal R\text{ for every } F \in \mathcal R\}$$ is a $\sigma$-algebra.

I can show that $S$ is closed under the countable union of it's subsets. But how I can show that if $E\in S$, then $E^c \in S$?

  • $\begingroup$ "its" not "it's" $\endgroup$ Commented May 25, 2015 at 11:18

1 Answer 1


Let: $E\in S$. For every $F\in \mathcal R$, we know that $E\cap F \in \mathcal R$. So, $F-(E\cap F) \in \mathcal R$. And it's easy to see that, $E^c \cap F=F-(E\cap F)$. Thus, $E^c \cap F \in \mathcal R$. So, $E^c\in S$.

  • $\begingroup$ Why is $F-(E\cap F)\in \mathcal R$? What is your definition of $\sigma$-ring? $\endgroup$ Commented May 25, 2015 at 16:16
  • $\begingroup$ Real Analysis by Folland (page 24-exercise 1) $\endgroup$
    – GhD
    Commented May 25, 2015 at 16:32
  • $\begingroup$ I'd prefer not to purchase a book to know your definition. There are multiple definitions, the most basic is "closed under countable union". $\endgroup$ Commented May 25, 2015 at 16:35
  • $\begingroup$ please give a reference for your definition from $\sigma-$ring. $\endgroup$ Commented May 25, 2015 at 16:42
  • $\begingroup$ Folland says a ring is closed under finite unions and differences. $\endgroup$
    – Mark
    Commented Sep 7, 2017 at 1:36

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