# Power Set is Sigma-Algebra

## Theorem

The power set of a set is a sigma-algebra.

## Proof

Let $S$ be a set, and let $\powerset S$ be its power set.

We have that a power set is an algebra of sets, and so:

$(1): \quad \forall A, B \in \powerset S: A \cup B \in \powerset S$
$(2): \quad \relcomp S A \in \powerset S$

Let $\sequence {A_i}$ be a countably infinite sequence of sets in $\powerset S$.

$\ds \bigcup_{i \mathop \in \N} A_i \in \powerset S$

So, by definition, $\powerset S$ is a sigma-algebra.

$\blacksquare$