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

The values $|f(x)|^{a}$ are nonnegative by Properties of Real Powers (ii). The function $|f|^{a}$ is the composition of $f$ with $u\mapsto|u|^{a}$, which is continuous on $\mathbb{R}$ by Properties of Real Powers (viii). Since $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-algebra of $(\mathbb{R},d_{\mathbb{R}})$ (Borel Sigma-Algebra of a Metric Space (ii)), this map is $\mathcal{B}(\mathbb{R})/\mathcal{B}(\mathbb{R})$-measurable by Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (iv), and so $|f|^{a}$ is measurable as a function into $\mathbb{R}$ by Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (ii). Being nonnegative, it is then measurable as a function into $[0,\infty]$ by Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (vii), that is, an element of $\mathcal{M}^{+}(X,\mathcal{F})$.