Proof of Integrable and Square-Integrable Vector-Valued Functions Are Maps into $\mathbb{R}^{n}$

Integrable functions. Let $f\in\mathcal{L}^{1}(\mu;\mathbb{R}^{n})$. By Integrable Vector- and Matrix-Valued Functions and Their Integrals (i), $f\in\mathrm{Meas}(X,\mathcal{F};\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}))$, whose elements are the measurable maps $X\to\mathbb{R}^{n}$ by Measurable Map, and the Set of Measurable Maps (ii) ($(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}))$ being a measurable space as recorded in Measurable Map, and the Set of Measurable Maps); in particular $f$ is a map $f:X\to\mathbb{R}^{n}$. This proves (i).

Square-integrable functions. Let $f\in\mathcal{L}^{2}(\mu;\mathbb{R}^{n})$. By Square-Integrable Vector-Valued Functions (ii), the elements of $\mathcal{L}^{2}(\mu;\mathbb{R}^{n})$ are measurable maps $X\to\mathbb{R}^{n}$; in particular $f$ is a map $f:X\to\mathbb{R}^{n}$. This proves (ii).

The probability case. Let $(\Omega,\mathcal{A},P)$ be a probability space and $Z\in\mathcal{L}^{2}(P;\mathbb{R}^{n})$. By Probability Spaces, Events and Random Elements (i), $(\Omega,\mathcal{A},P)$ is a measure space, so (ii), applied to the measure space $(\Omega,\mathcal{A},P)$ in place of $(X,\mathcal{F},\mu)$ and to $f=Z$, shows that $Z$ is a map $Z:\Omega\to\mathbb{R}^{n}$. This proves (iii).