Proof of Riesz Representation Theorem: the Dual of the Continuous Functions with Compact Support Is the Space of Finite Signed Borel Measures

Throughout, continuous real functions on $X$ are Borel measurable, and every $f\in C_{c}(X)$ satisfies $|f(x)|\le\|f\|_{u}$ for every $x\in X$; and $\nu\mapsto\Phi(\nu)$ is a function on $M(X)$, the condition $r=\int_{X}f\,d\nu$ being predicative in $(\nu,f,r)$ by The Simple, Nonnegative, Integrable and Signed Integrals Are Predicative in the Measure and the Function, at Any Variable Positions (iv) (and Maps and Relations Given by Formulas (i)).

The functional and the upper bound. Let $\nu\in M(X)$. For $f\in C_{c}(X)$, Properties of the Integral with Respect to a Finite Signed Measure: Linearity, Bounds, Bounded Functions, Linearity in the Measure, Indicators and Finite Measures (iii), with $C=\|f\|_{u}$, shows that $f$ is integrable with respect to $\nu$ and gives the bound below. The function $\Phi(\nu)$ is linear by Properties of the Integral with Respect to a Finite Signed Measure: Linearity, Bounds, Bounded Functions, Linearity in the Measure, Indicators and Finite Measures (i), and

$$|\Phi(\nu)(f)|\le\|\nu\|\,\|f\|_{u}\qquad(f\in C_{c}(X)).$$

So the nonnegative number $\|\nu\|$ is a bound for $\Phi(\nu)$; hence $\Phi(\nu)$ is bounded, $\Phi(\nu)\in C_{c}(X)^{*}$, and $\|\nu\|$ lies in the set of nonnegative bounds of $\Phi(\nu)$, whose infimum is $\|\Phi(\nu)\|_{*}$ by The Dual Norm (i). Thus $\|\Phi(\nu)\|_{*}\le\|\nu\|$.

Compact parts of a Hahn decomposition. Let $P$ be a Hahn set for $\nu$, which exists by Hahn and Jordan Decompositions of a Finite Signed Measure (i) (using the axiom of dependent choice). By Properties of the Total Variation: a Finite Measure, the Jordan Decomposition, Linear Combinations, the Total Variation Norm, and Integrability (ii) with $A=X$,

$$\|\nu\|=|\nu|(X)=\nu(P)-\nu(X\setminus P),$$

and $|\nu|$ is a finite Borel measure by Properties of the Total Variation: a Finite Measure, the Jordan Decomposition, Linear Combinations, the Total Variation Norm, and Integrability (i). Let $\varepsilon>0$. By Finite Borel Measures on a Locally Compact Separable Metric Space Are Radon (i), applied to $|\nu|$ and the Borel sets $P$ and $X\setminus P$, there are compact sets $K\subseteq P$ and $L\subseteq X\setminus P$ with $|\nu|(K)>|\nu|(P)-\varepsilon$ and $|\nu|(L)>|\nu|(X\setminus P)-\varepsilon$, that is,

$$|\nu|(P\setminus K)<\varepsilon,\qquad|\nu|\bigl((X\setminus P)\setminus L\bigr)<\varepsilon .$$

The sets $K$ and $L$ are disjoint.

A test function. The compact set $L$ is closed, so $X\setminus L$ is open and contains $K$, and Thickenings, Neighbourhoods with Compact Closure, Urysohn Functions and Partitions of Unity in a Locally Compact Metric Space (iii) gives $f_{1}\in C_{c}(X)$ with values in $[0,1]$, $f_{1}=1$ on $K$ and $\operatorname{supp}f_{1}\subseteq X\setminus L$. The compact set $\operatorname{supp}f_{1}$ is closed, so $X\setminus\operatorname{supp}f_{1}$ is open and contains $L$, and the same lemma gives $f_{2}\in C_{c}(X)$ with values in $[0,1]$, $f_{2}=1$ on $L$ and $\operatorname{supp}f_{2}\subseteq X\setminus\operatorname{supp}f_{1}$. Let $g=f_{1}-f_{2}\in C_{c}(X)$. Since the supports of $f_{1}$ and $f_{2}$ are disjoint, at every point at least one of $f_{1},f_{2}$ vanishes, by Continuous Functions with Compact Support: the Zero Function, Closure under Operations, Vanishing off the Support, Existence of the Maximum Modulus, and Uniform Continuity (iii). Hence $|g(x)|\le1$ for every $x\in X$, so $\|g\|_{u}\le1$; and $g=1$ on $K$, where $f_{1}=1\ne0$, and $g=-1$ on $L$, where $f_{2}=1\ne0$.

The lower bound. Let $h=\mathbf{1}_{P}-\mathbf{1}_{X\setminus P}$. By Properties of the Integral with Respect to a Finite Signed Measure: Linearity, Bounds, Bounded Functions, Linearity in the Measure, Indicators and Finite Measures (vi) and Properties of the Integral with Respect to a Finite Signed Measure: Linearity, Bounds, Bounded Functions, Linearity in the Measure, Indicators and Finite Measures (i), $h$ is integrable with respect to $\nu$ and

$$\int_{X}h\,d\nu=\nu(P)-\nu(X\setminus P)=\|\nu\|.$$

The function $g-h$ vanishes on $K\cup L$, and $|g-h|\le2$ everywhere, so

$$|g(x)-h(x)|\le2\cdot\mathbf{1}_{P\setminus K}(x)+2\cdot\mathbf{1}_{(X\setminus P)\setminus L}(x)\qquad(x\in X).$$

By Properties of the Integral with Respect to a Finite Signed Measure: Linearity, Bounds, Bounded Functions, Linearity in the Measure, Indicators and Finite Measures (i) and Properties of the Integral with Respect to a Finite Signed Measure: Linearity, Bounds, Bounded Functions, Linearity in the Measure, Indicators and Finite Measures (ii), and monotonicity of the integral with respect to the measure $|\nu|$,

$$\Bigl|\int_{X}g\,d\nu-\|\nu\|\Bigr|=\Bigl|\int_{X}(g-h)\,d\nu\Bigr|\le\int_{X}|g-h|\,d|\nu|\le2\,|\nu|(P\setminus K)+2\,|\nu|\bigl((X\setminus P)\setminus L\bigr)<4\varepsilon .$$

Since the dual norm is a bound for $\Phi(\nu)$,

$$\|\nu\|-4\varepsilon<\int_{X}g\,d\nu=\Phi(\nu)(g)\le\|\Phi(\nu)\|_{*}\,\|g\|_{u}\le\|\Phi(\nu)\|_{*}.$$

As $\varepsilon>0$ was arbitrary, $\|\nu\|\le\|\Phi(\nu)\|_{*}$. With the upper bound this proves (i).

Linearity. Let $\nu,\nu'\in M(X)$ and $c\in\mathbb{R}$; then $\nu+\nu'$ and $c\nu$ lie in $M(X)$ by Properties of the Total Variation: a Finite Measure, the Jordan Decomposition, Linear Combinations, the Total Variation Norm, and Integrability (v). For $f\in C_{c}(X)$, which is measurable and bounded by $\|f\|_{u}$, Properties of the Integral with Respect to a Finite Signed Measure: Linearity, Bounds, Bounded Functions, Linearity in the Measure, Indicators and Finite Measures (v) gives

$$\Phi(\nu+\nu')(f)=\Phi(\nu)(f)+\Phi(\nu')(f),\qquad\Phi(c\nu)(f)=c\,\Phi(\nu)(f).$$

Since the sum and the scalar multiples in $C_{c}(X)^{*}$ are pointwise (The Dual of a Real Normed Space Is a Real Banach Space; Weak-Star Limits (i)), this is (ii).

Existence of a representing measure. Let $\Lambda\in C_{c}(X)^{*}$. By Jordan Decomposition of a Bounded Linear Functional on the Continuous Functions with Compact Support (i) there are positive linear functionals $\Lambda^{+},\Lambda^{-}\in C_{c}(X)^{*}$ with $\Lambda f=\Lambda^{+}f-\Lambda^{-}f$ for every $f\in C_{c}(X)$. By The Riesz–Markov Representation Theorem on a Locally Compact Metric Space (i) there are Radon measures $\mu^{+}$ and $\mu^{-}$ on $X$ with

$$\Lambda^{+}f=\int_{X}f\,d\mu^{+},\qquad\Lambda^{-}f=\int_{X}f\,d\mu^{-}\qquad(f\in C_{c}(X)),$$

and by The Riesz–Markov Representation Theorem on a Locally Compact Metric Space (iii), $\mu^{\pm}(X)=\|\Lambda^{\pm}\|_{*}<\infty$ ($\mu^{\pm}$ is the measure of The Riesz–Markov Representation Theorem on a Locally Compact Metric Space (i) for $\Lambda^{\pm}$, unique by The Riesz–Markov Representation Theorem on a Locally Compact Metric Space (ii)), so $\mu^{+}$ and $\mu^{-}$ are finite Borel measures. Their difference $\nu$, $\nu(B)=\mu^{+}(B)-\mu^{-}(B)$ for $B\in\mathcal{B}(X)$, is a finite signed Borel measure, so $\nu\in M(X)$. For $f\in C_{c}(X)$, measurable and bounded by $\|f\|_{u}$, Properties of the Integral with Respect to a Finite Signed Measure: Linearity, Bounds, Bounded Functions, Linearity in the Measure, Indicators and Finite Measures (iv) gives

$$\int_{X}f\,d\nu=\int_{X}f\,d\mu^{+}-\int_{X}f\,d\mu^{-}=\Lambda^{+}f-\Lambda^{-}f=\Lambda f .$$

Thus $\Lambda=\Phi(\nu)$.

Uniqueness. Let $\nu,\nu'\in M(X)$ with $\Phi(\nu)=\Lambda=\Phi(\nu')$. Then $\int_{X}f\,d\nu=\int_{X}f\,d\nu'$ for every $f\in C_{c}(X)$, and since $X$ is locally compact and separable, Radon Measures and Finite Signed Borel Measures Are Determined by Their Integrals of Compactly Supported Continuous Functions (iii) gives $\nu(B)=\nu'(B)$ for every $B\in\mathcal{B}(X)$, that is, $\nu=\nu'$. This proves (iii).

Isomorphism. By (iii), every $\Lambda\in C_{c}(X)^{*}$ is $\Phi(\nu)$ for exactly one $\nu\in M(X)$, so $\Phi$ is a bijection of $M(X)$ onto $C_{c}(X)^{*}$. Let $\nu,\nu'\in M(X)$. The map $\Phi$ is linear ((ii)), so $\Phi(\nu)-\Phi(\nu')=\Phi(\nu-\nu')$, and by (i),

$$\|\Phi(\nu)-\Phi(\nu')\|_{*}=\|\Phi(\nu-\nu')\|_{*}=\|\nu-\nu'\| .$$

This proves (iv).