Expectation

In the setting of Measurable Functions and the Lebesgue Integral: Standing Notation, let $(\Omega,\mathcal{F},P)$ be a probability space and let $n,p,q\in\mathbb{N}$.

(i) Expectation of a function into $[0,\infty]$. For $X\in\mathcal{M}^{+}(\Omega,\mathcal{F})$, the expectation of $X$ is

$$\mathbb{E}X=\int_{\Omega}X\,dP\in[0,\infty].$$

(ii) Integrable random variables. A random variable $X$ is integrable if $X\in\mathcal{L}^{1}(P)$, and then its expectation is

$$\mathbb{E}X=\int_{\Omega}X\,dP\in\mathbb{R}$$

(for $X\ge0$ it agrees with (i), by Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (vi)).

(iii) Integrable random vectors. A random vector $X$ in $\mathbb{R}^{n}$ is integrable if $X\in\mathcal{L}^{1}(P;\mathbb{R}^{n})$, and then its expectation is the point

$$\mathbb{E}X=\int_{\Omega}X\,dP=(\mathbb{E}X_{1},\dots,\mathbb{E}X_{n})\in\mathbb{R}^{n}$$

(Integrable Vector- and Matrix-Valued Functions and Their Integrals (ii)).

(iv) Integrable matrix-valued functions. For $X\in\mathcal{L}^{1}(P;\mathbb{R}^{p\times q})$, the expectation of $X$ is the matrix

$$\mathbb{E}X=\int_{\Omega}X\,dP\in\mathbb{R}^{p\times q},\qquad(\mathbb{E}X)_{ij}=\mathbb{E}X_{ij}\quad(i\in[p],\ j\in[q])$$

(Integrable Vector- and Matrix-Valued Functions and Their Integrals (iv)).