Locally Compact Metric Spaces and Compactly Supported Functions: Standing Notation

(i) Base. This setting extends Separable Real Normed Spaces and Hahn–Banach: Notation and Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity: their conventions and notation are in force.

(ii) Metric spaces. For a metric space $(X,d)$, $B(x,r)$ and $\bar{B}(x,r)$ denote the open ball $B_{d}(x,r)$ and the closed ball $\bar{B}_{d}(x,r)$, and $\mathcal{T}_{d}$ the topology of the open sets of $(X,d)$, all as in the base. As for normed spaces in Separable Real Normed Spaces and Hahn–Banach: Notation (ii), open subsets of $X$ refer to $(X,d)$, and closed and compact subsets of $X$ to the topological space $(X,\mathcal{T}_{d})$. For $A\subseteq X$, $\operatorname{cl}A$ is the closure and $\operatorname{int}A$ the interior of $A$; for nonempty $A\subseteq X$ and $x\in X$, $\operatorname{dist}(x,A)$ is the distance from $x$ to $A$. Background: Closure and Interior in a Metric Space: Containments, the Closure is Closed, the Interior is Open, The Distance to a Subset of a Metric Space: Bounds, the Lipschitz Estimate, Continuity, and Vanishing Exactly on the Closure, Closed Subsets of Compact Sets Are Compact, and Compact Subsets of a Metric Space Are Closed and Bounded, The Continuous Image of a Compact Subset of a Metric Space is Compact, Extreme Value Theorem on a Compact Subset of a Metric Space, Heine-Cantor Theorem: Continuity on a Compact Metric Space or a Compact Subset Implies Uniform Continuity, The Finite Intersection Property of a Compact Set and The Open Sets of a Metric Space Form a Set, Given by a Predicative Formula.

(iii) Real functions. For a metric space $(X,d)$, a real function on $X$ is a function $f:X\to\mathbb{R}$, and its continuity refers to $(X,d)$ and $(\mathbb{R},d_{\mathbb{R}})$, as in the base. For real functions $f,g$ on $X$ and $c\in\mathbb{R}$, the sum $f+g$, the product $fg$, the scalar multiple $cf$ and the constant function $0$ are the pointwise operations and the constant map of the base, and $|f|$, $\max(f,g)$ and $\min(f,g)$ are the functions

$$x\mapsto|f(x)|,\qquad x\mapsto\max\{f(x),g(x)\},\qquad x\mapsto\min\{f(x),g(x)\}$$

on $X$, formed with the maximum and minimum of two real numbers (Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii), The Maximum and Minimum of Two Elements of a Total Order (i)).

(iv) Vector and normed spaces. Background on real vector spaces and real normed spaces: Membership and the Zero Law of a Real Vector Space in Flat Form, Rearranging Sums, Differences and Two-Term Combinations in a Real Vector Space, The Triangle Inequality of a Norm Read in the Type Real, Membership in the Line ℝx₀ of a Real Vector Space, Linear Functionals Respect Differences, and Coefficients Along a Nonzero Vector Are Unique, The Norm of a Negative, and the Norm Distance of a Translate and The Norm Distance from the Origin Is the Norm.

(v) Sequences. Sequences and subsequences are those of the base. For a sequence $(f_{m})$ in $\mathbb{R}^{X}$ and $x\in X$, $(f_{m}(x))_{m}$ is the sequence in $\mathbb{R}$ of Evaluating a Sequence of Maps at a Point Gives a Sequence, and Its Range (i). Background: Evaluating a Sequence of Maps at a Point Gives a Sequence, and Its Range.

(vi) Use. The background facts carried by (ii), (iv) and (v), and their elementary consequences, are used without further citation.