Power Set is Sigma-Algebra

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


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 $\powerset S$.

Then from Power Set is Closed under Countable Unions:

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

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