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Let $(\Omega,\mathscr A)$ be a measurable space. If $\varnothing \subset X \subset \Omega$, let $$\mathscr F = \{ F \subseteq \Omega, F = X \cap Y, Y \in \mathscr A\} \;. $$

I need to prove that $\mathscr F$ is a $ \sigma$-Algebra on $X$.

So, I have to show that

  1. $\varnothing \in \mathscr F$
  2. If $F \in \mathscr F$, then $F^C \in \mathscr F $
  3. If $F_i \in \mathscr F$, then $\bigcup_{i=1}^\infty F_i \in \mathscr F $

I have trouble in showing 2 and 3 conditions.

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    In condition 2, one asks that $X\setminus F\in\mathscr F$ for every $F\in\mathscr F$, not that $\Omega\setminus F\in\mathscr F$.2012-09-15
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    Yes you are right. New Sigma Algebra should be on X. Does it make us go further?2012-09-15
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    Yes: for example, you could try to write down $X\setminus F$ using $\Omega\setminus F$.2012-09-15
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    Ok, I am trying to understand this: So we have to show that $X \setminus F \in \mathscr F$. I can't figure out how to use Y in this case.2012-09-15
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    So... you assume that $F=X\cap Y$ with $Y$ in $\mathscr A$ and you want to find $Z$ in $\mathscr A$ such that $X\setminus F=X\cap Z$. Any idea?2012-09-15

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