Bounded Continuous Functions on a Metric Space: Measurability, Integrals, Operations, Lipschitz Functions and Approximation of Closed Sets

In the setting of Measurable Functions and the Lebesgue Integral: Standing Notation and Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity, let $(S,d)$ and $(T,\rho)$ be metric spaces with Borel $\sigma$-algebras $\mathcal{B}(S)$ and $\mathcal{B}(T)$, and let $C_{b}(S)$ and $C_{b}(T)$ be their sets of bounded continuous functions.

(i) Measurability. Every $f\in C_{b}(S)$ is $\mathcal{B}(S)/\mathcal{B}(\mathbb{R})$-measurable, with $\mathcal{B}(\mathbb{R})$ the Borel $\sigma$-algebra of $\mathbb{R}$.

(ii) Integrals. Let $\mu$ be a finite measure on $(S,\mathcal{B}(S))$, let $f\in C_{b}(S)$, and let $M\ge0$ be a real number with $|f(x)|\le M$ for every $x\in S$. Then $f\in\mathcal{L}^{1}(\mu)$ and

$$\Bigl|\int_{S}f\,d\mu\Bigr|\le M\,\mu(S).$$

(iii) Sums, multiples and products. Every constant function $S\to\mathbb{R}$ belongs to $C_{b}(S)$, and for $f,g\in C_{b}(S)$ and $a\in\mathbb{R}$ the functions $f+g$, $af$ and $fg$ belong to $C_{b}(S)$.

(iv) Composition on the left. For $f\in C_{b}(S)$ and a continuous function $\varphi:\mathbb{R}\to\mathbb{R}$, $\varphi\circ f\in C_{b}(S)$.

(v) Composition on the right. For $f\in C_{b}(S)$ and a continuous function $h:T\to S$, $f\circ h\in C_{b}(T)$.

(vi) Bounded Lipschitz functions. Every bounded Lipschitz function $f:S\to\mathbb{R}$ belongs to $C_{b}(S)$.

(vii) Approximation of closed sets. Let $F\subseteq S$ be nonempty and closed in $(S,\mathcal{T}_{d})$, with $\mathcal{T}_{d}$ the metric topology, and for $k\in\mathbb{N}$ let $f_{k}:S\to\mathbb{R}$ be the function

$$f_{k}(x)=\max\bigl\{0,\ 1-k\operatorname{dist}(x,F)\bigr\},$$

with $\operatorname{dist}(x,F)$ the distance from $x$ to $F$. Then each $f_{k}$ is Lipschitz with constant $k$, and for every $x\in S$ and $k\in\mathbb{N}$

$$0\le f_{k+1}(x)\le f_{k}(x)\le1,\qquad f_{k}(x)=1\ \text{ if }x\in F,\qquad f_{k}(x)\to\mathbf{1}_{F}(x)\quad(k\to\infty).$$