# Rudin functional analysis theorem 3.28, application of Reisz representation theorem.

Suppose

(a) $$X$$ is a topological vector space on which $$X^*$$ separates points,

(b) $$Q$$ is a compact subset of $$X$$, and

(c) the closed convex hull $$\overline{H}$$ of $$Q$$ is compact

then $$y \in \overline{H}$$ iff there's a regular Borel probability measure $$\mu$$ on $$Q$$ such that $$y = \int_Q x d \mu (x)$$

First part of the proof

Regard $$X$$ again as a real vector space. Let $$C(Q)$$ be the Banach space of all real continuous functions on $$Q$$, with the supremum norm. The Reisz representation theorem identifies the dual space $$C(Q)^*$$ with the space of all real Borel measure on $$Q$$ that are differences of regular positive ones.

I assume the Reisz representation theorem used is the following (Theorem 6.19 from Rudin's Real and Complex analysis).

If $$X$$ is a locally compact Hausdorff space, then every bounded linear functional $$\Phi$$ on $$C_0(X)$$ is represented by a unique regular complex Borel measure $$\mu$$, in the sense that $$\Phi f = \int_X f d \mu$$ for every $$f \in C_0(X)$$. Moreover, the norm of $$\Phi$$ is the total variation of $$\mu$$: $$\lVert \Phi \rVert = | \mu |(X)$$

The bit I don't get in the proof is "Borel measures on $$Q$$ that are differences of regular positive ones".

Where does the "differences" come from?