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WeeG
04-08-2010, 03:23 AM
hi, I need some help with this one....

Let \Omega be a finite set. Show that the set of all subsets of \Omega (2^omega), is also finite and that it is a \sigma algebra.

thanks !

Martingale
04-08-2010, 12:41 PM
hi, I need some help with this one....

Let \Omega be a finite set. Show that the set of all subsets of \Omega (2^omega), is also finite and that it is a \sigma algebra.

thanks !

if |\Omega|=n<\infty then |\mathcal{P}(\Omega)|=2^n which is finite.

As for the \sigma-algebra part.

\mathcal{P}(\Omega) is nonempty since it contains the empty set.

If we let X\in\mathcal{P}(\Omega) then X^c=\Omega\backslash X\subset\Omega. and since \mathcal{P}(\Omega) contains all subsets of \Omega we have X^c\in\mathcal{P}(\Omega)

similarly for countable unions