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 $\powerset S$.

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.

$\blacksquare$

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $3.3 \ \text{(i)}$
• 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $1.1$: Algebras and Sigma-Algebras