The Metric Subspace: Continuity, the Borel Sigma-Algebra, and the Restriction of a Borel Measure

In the setting of Measure Spaces and the Lebesgue Integral: Standing Notation, whose measure space is instantiated at each use below by the measure space named there, so that the letter $X$ is free for the use fixed here: let $(X,d)$ be a metric space, let $\mathcal{T}_{d}$ be the collection of subsets of $X$ that are open in $(X,d)$, which is a topology on $X$ by Metric Open Sets Form a Topology, and let $\mathcal{B}(X)$ be the Borel $\sigma$-algebra of $(X,d)$.

Let $A\subseteq X$, let $d_{A}$ be the restriction of $d$ to $A$, which is a metric on $A$ by claim 1 of that lemma, and let $\mathcal{B}(A)$ be the Borel $\sigma$-algebra of the metric space $(A,d_{A})$. Let $\mathbb{R}$ denote the real numbers, regarded as a metric space through the absolute-value metric, and let $\mathcal{B}(\mathbb{R})$ be the Borel $\sigma$-algebra of the real line. Measurability of a real-valued or $[0,\infty]$-valued function on a measurable space is that of Measure Spaces and the Lebesgue Integral: Standing Notation §measurable.

Then the following hold.

1. (Continuity and a subspace)¶ Let $(Y,d_{Y})$ be a metric space.

(a)¶ Let $f:A\to Y$. Then $f$ is continuous on $A$ as a map from $(X,d)$ to $(Y,d_{Y})$ if and only if $f$ is continuous on $A$ as a map from the metric space $(A,d_{A})$ to $(Y,d_{Y})$.

(b)¶ Let $C\subseteq Y$, let $d_{C}$ be the restriction of $d_{Y}$ to $C$, which is a metric on $C$ by claim 1 of The Restriction of a Metric to a Subset Induces the Subspace Topology read for the metric space $(Y,d_{Y})$ and the subset $C$, let $u:A\to Y$ satisfy $u(x)\in C$ for every $x\in A$, and let $u_{C}:A\to C$ be the map with $u_{C}(x)=u(x)$ for every $x\in A$. Then $u$ is continuous on $A$ as a map into $(Y,d_{Y})$ if and only if $u_{C}$ is continuous on $A$ as a map into $(C,d_{C})$.

2. (Trace of the Borel sets)¶ $\mathcal{B}(A)=\{B\cap A:B\in\mathcal{B}(X)\}$.

3. (A Borel subspace, and the restriction of a measure)¶ Assume that $A\in\mathcal{B}(X)$. Then

which is the $\sigma$-algebra $\mathcal{B}(X)|_{A}$ of claim 1 of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions, read for the measurable space $(X,\mathcal{B}(X))$ and the set $A$. Consequently, for every Borel measure $\mu$ on $(X,d)$, the restriction $\mu|_{A}$ of that claim is a Borel measure on $(A,d_{A})$, with $\mu|_{A}(B)=\mu(B)$ for every $B\in\mathcal{B}(A)$; in particular $\mu|_{A}(A)=\mu(A)$.

4. (Integrating a function that vanishes off $A$)¶ Assume that $A\in\mathcal{B}(X)$, let $\mu$ be a Borel measure on $(X,d)$, and let $f:X\to\mathbb{R}$ be measurable with respect to $\mathcal{B}(X)$ and $\mathcal{B}(\mathbb{R})$, integrable with respect to $\mu$, and such that $f(x)=0$ for every $x\in X\setminus A$. Then the restriction $f|_{A}$ is measurable with respect to $\mathcal{B}(A)$ and $\mathcal{B}(\mathbb{R})$, is integrable with respect to $\mu|_{A}$, and