Square-Integrable Vector-Valued Functions: Components, Dot Products and Almost-Everywhere Equality

In the setting of Measurable Functions and the Lebesgue Integral: Standing Notation, let $(X,\mathcal{F},\mu)$ be a measure space and let $n\in\mathbb{N}$. For measurable $f:X\to\mathbb{R}^{n}$, $|f|^{2}$ is the function of Square-Integrable Vector-Valued Functions (i), and $\mathcal{L}^{2}(\mu;\mathbb{R}^{n})$ is the set of square-integrable functions from $X$ to $\mathbb{R}^{n}$. $\mathcal{L}^{2}(\mu)$ is the set of $2$-integrable functions $X\to\mathbb{R}$, and $\|g\|_{2}$, for $g\in\mathcal{L}^{2}(\mu)$, is the $2$-seminorm. For functions $f,g:X\to\mathbb{R}^{n}$ and $a,b\in\mathbb{R}$, $af+bg$, $f+g$, $af$, $-f$ and $f-g$ are the functions $x\mapsto af(x)+bg(x)$, $x\mapsto f(x)+g(x)$, $x\mapsto af(x)$, $x\mapsto-f(x)$ and $x\mapsto f(x)-g(x)$, and $f\cdot g$ is the function $x\mapsto f(x)\cdot g(x)$, with the dot product; integrals of $f\cdot g$ and of $f_{k}g_{k}$ are taken in $\mathcal{L}^{1}(\mu)$. For $f,g\in\mathcal{L}^{2}(\mu;\mathbb{R}^{n})$, $f\sim g$ means that $f(x)=g(x)$ for $\mu$-almost every $x\in X$; $\sim$ is this relation on $\mathcal{L}^{2}(\mu;\mathbb{R}^{n})$.

(i) Components. A function $f:X\to\mathbb{R}^{n}$ belongs to $\mathcal{L}^{2}(\mu;\mathbb{R}^{n})$ if and only if its component $f_{k}$ belongs to $\mathcal{L}^{2}(\mu)$ for every $k\in[n]$. In that case

$$\int|f|^{2}\,d\mu=\sum_{k=1}^{n}\|f_{k}\|_{2}^{2}.$$

(ii) Dot products. Let $f,g\in\mathcal{L}^{2}(\mu;\mathbb{R}^{n})$. Then $f\cdot g\in\mathcal{L}^{1}(\mu)$ and

$$\int f\cdot g\,d\mu=\sum_{k=1}^{n}\int f_{k}g_{k}\,d\mu .$$

(iii) Linear combinations. Let $f,g\in\mathcal{L}^{2}(\mu;\mathbb{R}^{n})$ and $a,b\in\mathbb{R}$. Then $af+bg\in\mathcal{L}^{2}(\mu;\mathbb{R}^{n})$; in particular $f+g$, $af$, $-f$ and $f-g$ belong to $\mathcal{L}^{2}(\mu;\mathbb{R}^{n})$.

(iv) Almost-everywhere equality. $\sim$ is an equivalence relation on $\mathcal{L}^{2}(\mu;\mathbb{R}^{n})$. For $f,g\in\mathcal{L}^{2}(\mu;\mathbb{R}^{n})$, $f\sim g$ if and only if, for every $k\in[n]$, $f_{k}(x)=g_{k}(x)$ for $\mu$-almost every $x\in X$.

(v) Compatibility. Let $f,f',g,g'\in\mathcal{L}^{2}(\mu;\mathbb{R}^{n})$ with $f\sim f'$ and $g\sim g'$, and let $a\in\mathbb{R}$. Then $f+g\sim f'+g'$, $af\sim af'$ and

$$\int f\cdot g\,d\mu=\int f'\cdot g'\,d\mu .$$