# Show that power set is a sigma-algebra

• Nov 20th 2010, 05:55 AM
ecnanif
Show that power set is a sigma-algebra
Hi

Let $\displaystyle \Omega$ be a finite set. Show that the set of all subsets of $\displaystyle \Omega$, $\displaystyle 2^{\Omega}$, is also finite and that it is a $\displaystyle \sigma$-algebra.

Solution (?)

Since $\displaystyle \Omega$ is finite, $\displaystyle \exists$ $\displaystyle n \in \mathbb{N}$ such that $\displaystyle card(\Omega) \leq n$. From set theory we know that the number of subsets of a finite set of cardinality $\displaystyle n$ is $\displaystyle 2^{n}$. Therefore,
$\displaystyle card(2^{\Omega}) \leq 2^{n} < \infty$, since $\displaystyle n < \infty$. Hence, $\displaystyle 2^{\Omega}$ is finite.
$\displaystyle \left\{\emptyset\right\}$ and $\displaystyle \left\{\Omega\right\}$ is in $\displaystyle 2^{\Omega}$. Since $\displaystyle 2^{\Omega}$ contains all subsets of $\displaystyle \Omega$ it must also contain all necessary unions, intersections and complements to make $\displaystyle 2^{\Omega}$ a $\displaystyle \sigma$-algebra.

Thanks!
• Nov 20th 2010, 06:06 AM
FernandoRevilla
Quote:

Originally Posted by ecnanif
$\displaystyle \left\{\emptyset\right\}$ and $\displaystyle \left\{\Omega\right\}$ is in $\displaystyle 2^{\Omega}$

You should say $\displaystyle \emptyset,\;\Omega$ instead of $\displaystyle \left\{{\emptyset}\right\},\;\left\{{\Omega}\right \}$.

Regards.

Fernando Revilla
• Nov 20th 2010, 06:08 AM
ecnanif
Quote:

Originally Posted by FernandoRevilla
You should say $\displaystyle \emptyset,\;\Omega$ instead of $\displaystyle \left\{{\emptyset}\right\},\;\left\{{\Omega}\right \}$.

Regards.

Fernando Revilla

Ok, but besides from this, correct?
• Nov 20th 2010, 06:10 AM
FernandoRevilla
Quote:

Originally Posted by ecnanif
Ok, but besides from this, correct?

Yes.

Regards.

Fernando Revilla