Function Measurable with respect to Power Set

Theorem

Let $\struct {X, \map \PP X}$ be a measurable space, where $\map \PP X$ is the power set of $X$.

Let $f : X \to \overline \R$ be a function.

Then $f$ is $\map \PP X$-measurable function.

Proof

For each $\alpha \in \R$, we have:

$\set {x \in X : \map f x \le \alpha} \subseteq X$

That is, from the definition of power set:

$\set {x \in X : \map f x \le \alpha} \in \map \PP X$

So for each $\alpha \in \R$:

the set $\set {x \in X : \map f x \le \alpha}$ is $\map \PP X$-measurable.

So:

$f$ is $\map \PP X$-measurable.

$\blacksquare$