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:


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

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

Consider an element of the union of all the sets in this sequence:
 * $\displaystyle x \in \bigcup_{i \mathop \in \N} A_i$

By definition of union:
 * $\exists i \in \N: x \in A_i$

But $A_i \in \powerset S$ and so by definition $A_i \subseteq S$.

By definition of subset, it follows that $x \in S$.

Hence, again by definition of subset:
 * $\displaystyle \bigcup_{i \mathop \in \N} A_i \subseteq S$

Finally, by definition of power set:
 * $\displaystyle \bigcup_{i \mathop \in \N} A_i \in \powerset S$

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