Sigma-Algebra

**MrJack1990** · Aug 7th 2009, 07:01 PM

Let $A,B,C$ be a partition of $\Omega$ . Write out the smallest $\sigma$ -algebra containing the sets $A,B,$ and $C$

**MrJack1990** · Aug 7th 2009, 07:43 PM

I have no idea if this is correct, but this is my best estimation of what the question is asking.

A partition is such that
$A \cap B = \emptyset$
$A \cap C = \emptyset$
$B \cap C = \emptyset$

and $A \cup B \cup C = \Omega$ i.e. is exhaustive

Then by definition of borel-algebra we have
(1) $\Omega \ \in \ \mathcal{F}$
(2) As $A,B,C \ \in \mathcal{F}, A^{c}, B^{c}, C^{c} \in \mathcal{F}$
(3) As $A,B,C \in \mathcal{F}, A \cup B \cup C \in \mathcal{F}$

Is this right?

**Moo** · Aug 7th 2009, 11:18 PM

Hello,

(you're asked for a sigma algebra, not a Borel algebra

)

So yup, the empty set and $\Omega$ belong to $\mathcal{F}$ (first+second axiom)

A,B,C belong to $\mathcal{F}$

Their complement belong to $\mathcal{F}$ .
But... you can have a more precise formula for these complements :

$A^c=B\cup C$
$B^c=A\cup C$
$C^c=A\cup B$

Do you agree ?
We don't need to include $A\cup B\cup C$ because it's $\Omega$ ...

And thus the third axiom is automatically checked.

So the smallest $\sigma$ -algebra containing A,B,C (also called the generated $\sigma$ -algebra by A,B,C) is :
$\mathcal{F}=\{\emptyset,A,B,C,A\cup B,A\cup C,B\cup C,\Omega\}$

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