Borel Measurability and Bounded Integration on a Metric Space

Let $(X,d)$ be a metric space, let $\mathcal{T}_d$ be its topology of open subsets, and let $\mathcal{B}(X)$ be its Borel $\sigma$-algebra. Measurability of maps between measurable spaces is that of Measurable Function and Real-Valued Measurable Function, and $\mathcal{B}(\mathbb{R})$ denotes the Borel $\sigma$-algebra of the real line.

1. (Open sets, closed sets, and generation) Every open subset of $(X,d)$ and every closed subset of $(X,\mathcal{T}_d)$ belongs to $\mathcal{B}(X)$. Moreover the family $\mathcal{C}$ of all closed subsets of $(X,\mathcal{T}_d)$ is a $\pi$-system whose generated $\sigma$-algebra is $\mathcal{B}(X)$.

2. (The real line) Let $d_{\mathbb{R}}$ be the absolute-value metric on $\mathbb{R}$, given by $d_{\mathbb{R}}(s,t)=|s-t|$. Then the Borel $\sigma$-algebra of the metric space $(\mathbb{R},d_{\mathbb{R}})$ is $\mathcal{B}(\mathbb{R})$.

3. (Continuous maps) Let $(Y,d_Y)$ be a metric space and let $f:X\to Y$ be continuous on $X$. Then $f$ is measurable with respect to $\mathcal{B}(X)$ and $\mathcal{B}(Y)$. In particular, if $A\subseteq X$ is nonempty then the function $x\mapsto\operatorname{dist}_d(x,A)$, the distance to $A$, is measurable with respect to $\mathcal{B}(X)$ and $\mathcal{B}(\mathbb{R})$.

4. (Composition) Let $(\Omega,\mathcal{F})$ and $(Z,\mathcal{G})$ be measurable spaces, let $Y:\Omega\to X$ be measurable with respect to $\mathcal{F}$ and $\mathcal{B}(X)$, and let $g:X\to Z$ be measurable with respect to $\mathcal{B}(X)$ and $\mathcal{G}$. Then $g\circ Y$ is measurable with respect to $\mathcal{F}$ and $\mathcal{G}$.

5. (Semicontinuous functions) Let $u:X\to\mathbb{R}$ be lower semicontinuous on $X$, or upper semicontinuous on $X$. Then $u$ is measurable with respect to $\mathcal{B}(X)$ and $\mathcal{B}(\mathbb{R})$.

6. (Bounded integration) Let $\mu$ be a Borel measure on $(X,d)$ with $\mu(X)<\infty$. Then:

(a) for every real $c$ the constant function on $X$ with value $c$ is integrable with respect to $\mu$, with $\int_X c\,d\mu=c\,\mu(X)$;

(b) if $f:X\to\mathbb{R}$ is measurable with respect to $\mathcal{B}(X)$ and $\mathcal{B}(\mathbb{R})$ and there is a real $M\ge0$ with $|f(x)|\le M$ for every $x\in X$, then $f$ is integrable with respect to $\mu$ and

$$\Bigl|\int_X f\,d\mu\Bigr|\le M\,\mu(X);$$

(c) if in addition $0\le f(x)$ for every $x\in X$, then the integral of $f$ as a nonnegative measurable function, in the sense of Lebesgue Integral of a Nonnegative Measurable Function, is real and equals $\int_X f\,d\mu$.