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.