Proof of Helly Selection Theorem for Finite Signed Borel Measures on a Locally Compact Separable Metric Space

Throughout, $\Phi$ is the function $\nu\mapsto\Phi(\nu)$ of Riesz Representation Theorem: the Dual of the Continuous Functions with Compact Support Is the Space of Finite Signed Borel Measures (i), $\Phi(\nu)(f)=\int_{X}f\,d\nu$, a function on $M(X)$ with its vector-space operations of Properties of the Total Variation: a Finite Measure, the Jordan Decomposition, Linear Combinations, the Total Variation Norm, and Integrability (vii) (Riesz Representation Theorem: the Dual of the Continuous Functions with Compact Support Is the Space of Finite Signed Borel Measures (iv)), and $B^{*}_{C}\subseteq C_{c}(X)^{*}$ is the ball of Weak-Star Compactness and Vague Convergence: Standing Notation (ii) for the normed space $(C_{c}(X),\|\cdot\|_{u})$.

The functionals of the measures. By Riesz Representation Theorem: the Dual of the Continuous Functions with Compact Support Is the Space of Finite Signed Borel Measures (i), $\Lambda_{m}=\Phi(\nu_{m})$ lies in $C_{c}(X)^{*}$ and $\|\Lambda_{m}\|_{*}=\|\nu_{m}\|\le C$, so $(\Lambda_{m})$ is a sequence in $B^{*}_{C}$.

A weak-star convergent subsequence. The normed space $(C_{c}(X),\|\cdot\|_{u})$ is separable by The Space of Continuous Functions with Compact Support on a Locally Compact Separable Metric Space is Separable (i). Hence Banach–Alaoglu Theorem for a Separable Normed Space: the Closed Balls of the Dual Are Weak-Star Sequentially Compact and Compact for a Metric of Weak-Star Convergence (i) gives a strictly increasing sequence $(m_{j})$ in $\mathbb{N}$ and a $\Lambda\in B^{*}_{C}$ such that $(\Lambda_{m_{j}})$ converges weak-star to $\Lambda$.

The limit is the functional of a measure. By Riesz Representation Theorem: the Dual of the Continuous Functions with Compact Support Is the Space of Finite Signed Borel Measures (iii) there is $\nu\in M(X)$ with $\Lambda=\Phi(\nu)$, and by Riesz Representation Theorem: the Dual of the Continuous Functions with Compact Support Is the Space of Finite Signed Borel Measures (i), $\|\nu\|=\|\Phi(\nu)\|_{*}=\|\Lambda\|_{*}\le C$. Weak-star convergence of $(\Lambda_{m_{j}})$ to $\Lambda$ says that for every $f\in C_{c}(X)$

$$\int_{X}f\,d\nu_{m_{j}}=\Lambda_{m_{j}}f\to\Lambda f=\int_{X}f\,d\nu,$$

which is the vague convergence of the subsequence $(\nu_{m_{j}})$ to $\nu$. This proves (i).

Nonnegative measures: the limit functional is positive. Suppose now that $\nu_{m}(A)\ge0$ for every $m$ and every $A\in\mathcal{B}(X)$, and let $(\nu_{n_{j}})$ be any subsequence converging vaguely to some $\nu\in M(X)$ (in these last two paragraphs $\nu$ denotes this limit). Let $f\in C_{c}(X)$ with $f\ge0$. By Properties of the Integral with Respect to a Finite Signed Measure: Linearity, Bounds, Bounded Functions, Linearity in the Measure, Indicators and Finite Measures (vii), each $\nu_{n_{j}}$ is a finite measure and $\int_{X}f\,d\nu_{n_{j}}$ is the Lebesgue integral of the nonnegative function $f$, hence nonnegative. Passing to the limit, $\Phi(\nu)(f)=\int_{X}f\,d\nu\ge0$ by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (x). Thus $\Phi(\nu)$, a linear functional on $C_{c}(X)$ since it lies in $C_{c}(X)^{*}$, is positive.

Nonnegative measures: identification of the limit. By The Riesz–Markov Representation Theorem on a Locally Compact Metric Space (i) there is a Radon measure $\mu$ on $X$ with $\Phi(\nu)(f)=\int_{X}f\,d\mu$ for every $f\in C_{c}(X)$, and by The Riesz–Markov Representation Theorem on a Locally Compact Metric Space (iii), applied to the bounded functional $\Phi(\nu)$, $\mu$ is finite. So $\mu$ is a finite Borel measure on $X$, hence $\mu\in M(X)$ (a finite measure is a finite signed measure) with $\mu(A)\ge0$ for every $A\in\mathcal{B}(X)$, and by Properties of the Integral with Respect to a Finite Signed Measure: Linearity, Bounds, Bounded Functions, Linearity in the Measure, Indicators and Finite Measures (vii) its integrals as an element of $M(X)$ are its Lebesgue integrals. Hence $\Phi(\mu)(f)=\int_{X}f\,d\mu=\Phi(\nu)(f)$ for every $f\in C_{c}(X)$, that is, $\Phi(\mu)=\Phi(\nu)$. The uniqueness in Riesz Representation Theorem: the Dual of the Continuous Functions with Compact Support Is the Space of Finite Signed Borel Measures (iii) gives $\mu=\nu$, and therefore $\nu(A)=\mu(A)\ge0$ for every $A\in\mathcal{B}(X)$. This proves (ii).