Power Set is Sigma-Algebra

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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.

$\blacksquare$


Sources