Expectations Computed from the Law

In the setting of Probability Spaces: Notation, let $(S,d)$ be a metric space with Borel $\sigma$-algebra $\mathcal{B}(S)$, let $X$ be a random element of $S$, and let $P_{X}$ be its law.

(i) Nonnegative functions. Let $g:S\to[0,\infty]$ be measurable. Then $g(X)\in\mathcal{M}^{+}(\Omega,\mathcal{F})$ and

$$\mathbb{E}\,g(X)=\int_{S}g\,dP_{X}.$$

(ii) Real functions. Let $g:S\to\mathbb{R}$ be measurable. Then $g(X)$ is a random variable; it is integrable if and only if $g\in\mathcal{L}^{1}(P_{X})$, and in that case

$$\mathbb{E}\,g(X)=\int_{S}g\,dP_{X}.$$

(iii) Vector-valued functions. Let $m\in\mathbb{N}$ and let $g:S\to\mathbb{R}^{m}$ be measurable. Then $g(X)$ is a random vector in $\mathbb{R}^{m}$; it is integrable if and only if $g\in\mathcal{L}^{1}(P_{X};\mathbb{R}^{m})$, and in that case

$$\mathbb{E}\,g(X)=\int_{S}g\,dP_{X}.$$