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

In the setting of Measurable Functions and the Lebesgue Integral: Standing Notation, let $(X,\mathcal{F},\mu)$ be a measure space and let $n\in\mathbb{N}$.

(i) Every integrable function $f\in\mathcal{L}^{1}(\mu;\mathbb{R}^{n})$ is a map $f:X\to\mathbb{R}^{n}$.

(ii) Every square-integrable function $f\in\mathcal{L}^{2}(\mu;\mathbb{R}^{n})$ is a map $f:X\to\mathbb{R}^{n}$.

(iii) If $(\Omega,\mathcal{A},P)$ is a probability space, then every $Z\in\mathcal{L}^{2}(P;\mathbb{R}^{n})$ is a map $Z:\Omega\to\mathbb{R}^{n}$.