Real Powers of the Absolute Value of a Measurable Function Are Measurable

In the setting of Measurable Functions and the Lebesgue Integral: Standing Notation, let $(X,\mathcal{F})$ be a measurable space. For real $t\ge0$ and $a>0$, $t^{a}$ is the real power of Real Powers (i) and Real Powers (ii).

(i) Measurability. Let $f:X\to\mathbb{R}$ be measurable and let $a>0$ be a real number. Then the function

$$|f|^{a}:X\to\mathbb{R},\qquad x\mapsto|f(x)|^{a},$$

is measurable; its values are nonnegative, and it is an element of $\mathcal{M}^{+}(X,\mathcal{F})$.