Radon Measures on Locally Compact Metric Spaces: Standing Notation

(i) Base. This setting extends Locally Compact Metric Spaces and Compactly Supported Functions: Standing Notation and Measurable Functions and the Lebesgue Integral: Standing Notation: their conventions and notation are in force.

(ii) Borel measures. For a metric space $(X,d)$, $\mathcal{B}(X)$ is its Borel $\sigma$-algebra, as in the base. A Borel measure on $X$ means a measure on the measurable space $(X,\mathcal{B}(X))$, and a finite signed Borel measure on $X$ means a finite signed measure on $(X,\mathcal{B}(X))$.

(iii) Finite signed measures. Let $(X,\mathcal{F})$ be a measurable space. $M(X,\mathcal{F})$ denotes the set of finite signed measures on $(X,\mathcal{F})$. For $\nu,\nu'\in M(X,\mathcal{F})$ and $c\in\mathbb{R}$, $\nu+\nu'$ and $c\nu$ are formed by the pointwise operations on $\mathbb{R}^{\mathcal{F}}$, the set of functions $\mathcal{F}\to\mathbb{R}$:

$$(\nu+\nu')(A)=\nu(A)+\nu'(A),\qquad(c\nu)(A)=c\,\nu(A)\qquad(A\in\mathcal{F}).$$

For a finite signed measure $\nu$ on $(X,\mathcal{F})$ and a Hahn set $P$ for $\nu$, $\nu^{+}$ and $\nu^{-}$ are the finite measures of Hahn and Jordan Decompositions of a Finite Signed Measure (ii),

$$\nu^{+}(A)=\nu(A\cap P),\qquad\nu^{-}(A)=-\nu(A\setminus P)\qquad(A\in\mathcal{F}),$$

the Jordan decomposition of $\nu$. Background: Hahn and Jordan Decompositions of a Finite Signed Measure and Finite Signed Measures from Finite Measures and from Integrals.

(iv) Dependent choice. The axiom of dependent choice is a standing hypothesis of the items that adopt this setting, in place of the contrary remark of Measurable Functions and the Lebesgue Integral: Standing Notation (vii).

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