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?
You are using the wrong version of Riesz Representation Theorem In the chapter on Complex Measures there is a section on Riesz Representation Theorem where Rudin proves a more general version. You have to use that version.