Integrable Vector- and Matrix-Valued Functions and Their Integrals

In the settings of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation and Euclidean Space and Real Matrices: Standing Notation, let $(X,\mathcal{F},\mu)$ be a measure space and let $n,p,q\in\mathbb{N}$, with $\mathbb{R}^{p\times q}=\mathcal{M}_{p\times q}(\mathbb{R})$ as in Borel Sigma-Algebra of a Metric Space (iv). $\mathcal{L}^{1}(\mu)$ is the set of integrable functions $X\to\mathbb{R}$, with the integral $\int_{X}g\,d\mu\in\mathbb{R}$ of $g\in\mathcal{L}^{1}(\mu)$. For $f:X\to\mathbb{R}^{n}$ and $k\in[n]$, $f_{k}$ is the $k$-th component function of $f$; for $F:X\to\mathbb{R}^{p\times q}$, $i\in[p]$ and $j\in[q]$, $F_{ij}$ is its entry function. For a measurable space $(Y,\mathcal{G})$, $\mathrm{Meas}(X,\mathcal{F};Y,\mathcal{G})$ is the set of $\mathcal{F}/\mathcal{G}$-measurable functions $X\to Y$ of Measurable Map, and the Set of Measurable Maps (ii).

(i) Integrable functions into $\mathbb{R}^{n}$. $\mathcal{L}^{1}(X,\mathcal{F},\mu;\mathbb{R}^{n})$, also written $\mathcal{L}^{1}(\mu;\mathbb{R}^{n})$, is the set of those $f\in\mathrm{Meas}(X,\mathcal{F};\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}))$ with

$$f_{k}\in\mathcal{L}^{1}(\mu)\qquad(k\in[n])$$

(it exists by Sets and Functions Defined Componentwise on Families of Vector- and Matrix-Valued Functions (i)). Its elements are the integrable functions into $\mathbb{R}^{n}$ with respect to $\mu$: the measurable functions $f:X\to\mathbb{R}^{n}$ whose components all lie in $\mathcal{L}^{1}(\mu)$.

(ii) Integrals of functions into $\mathbb{R}^{n}$. For $f\in\mathcal{L}^{1}(\mu;\mathbb{R}^{n})$, the integral of $f$ with respect to $\mu$ is the point of $\mathbb{R}^{n}$

$$\int_{X}f\,d\mu=\Bigl(\int_{X}f_{1}\,d\mu,\dots,\int_{X}f_{n}\,d\mu\Bigr),\qquad\text{that is,}\qquad\Bigl(\int_{X}f\,d\mu\Bigr)_{k}=\int_{X}f_{k}\,d\mu\quad(k\in[n]),$$

also written $\int f\,d\mu$ (a point of $\mathbb{R}^{n}$ by Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (ii)); the function $f\mapsto\int_{X}f\,d\mu$ from $\mathcal{L}^{1}(\mu;\mathbb{R}^{n})$ to $\mathbb{R}^{n}$ is the integral with respect to $\mu$ (it exists by Sets and Functions Defined Componentwise on Families of Vector- and Matrix-Valued Functions (ii) with $\Phi$ the integral on $\mathcal{L}^{1}(\mu)$, as $\mathbb{R}^{[n]}=\mathbb{R}^{n}$ by Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (i)).

(iii) Integrable functions into $\mathbb{R}^{p\times q}$. $\mathcal{L}^{1}(X,\mathcal{F},\mu;\mathbb{R}^{p\times q})$, also written $\mathcal{L}^{1}(\mu;\mathbb{R}^{p\times q})$, is the set of those $F\in\mathrm{Meas}(X,\mathcal{F};\mathbb{R}^{p\times q},\mathcal{B}(\mathbb{R}^{p\times q}))$ with

$$F_{ij}\in\mathcal{L}^{1}(\mu)\qquad(i\in[p],\ j\in[q])$$

(it exists by Sets and Functions Defined Componentwise on Families of Vector- and Matrix-Valued Functions (iii)). Its elements are the integrable functions into $\mathbb{R}^{p\times q}$ with respect to $\mu$: the measurable functions $F:X\to\mathbb{R}^{p\times q}$ whose entries all lie in $\mathcal{L}^{1}(\mu)$.

(iv) Integrals of functions into $\mathbb{R}^{p\times q}$. For $F\in\mathcal{L}^{1}(\mu;\mathbb{R}^{p\times q})$, the integral of $F$ with respect to $\mu$ is the matrix $\int_{X}F\,d\mu\in\mathbb{R}^{p\times q}$ with entries

$$\Bigl(\int_{X}F\,d\mu\Bigr)_{ij}=\int_{X}F_{ij}\,d\mu\qquad(i\in[p],\ j\in[q]),$$

also written $\int F\,d\mu$ (a matrix by Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (i)); the function $F\mapsto\int_{X}F\,d\mu$ from $\mathcal{L}^{1}(\mu;\mathbb{R}^{p\times q})$ to $\mathbb{R}^{p\times q}$ is the integral with respect to $\mu$ (it exists by Sets and Functions Defined Componentwise on Families of Vector- and Matrix-Valued Functions (iv) with $\Phi$ the integral on $\mathcal{L}^{1}(\mu)$, as $\mathbb{R}^{[p]\times[q]}=\mathcal{M}_{p\times q}(\mathbb{R})$ by Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (i)).