Use Problem 7.7 to show that a function \(f: \mathbb{R}^{n} ightarrow \mathbb{R}^{m}, x \mapsto\left(f_{1}(x), \ldots, f_{m}(x)ight)\) is measurable if, and only if, all coordinate maps \(f_{i}: \mathbb{R}^{n} ightarrow \mathbb{R}, i=1,2, \ldots, m\), are measurable.

[ show that the coordinate projections \(x=\left(x_{1}, \ldots, x_{n}ight) \mapsto x_{i}\) are measurable.]

Data from problem 7.7

Let \(X\) be a set, let \(\left(X_{i}, \mathscr{A}_{i}ight), i \in I\), be arbitrarily many measurable spaces and let \(T_{i}: X ightarrow X_{i}\) be a family of maps. Show that a map \(f\) from a measurable space \((F, \mathscr{F})\) to \(\left(X, \sigma\left(T_{i}: i \in Iight)ight)\) is measurable if, and only if, all maps \(T_{i} \circ f\) are \(\mathscr{F} / \mathscr{A}_{i}\)-measurable.