Few bits a bit unclear of
a) $X$ is a topological vector space on which $X^*$ separates points, and
b) $\mu$ is a Borel probability measure on a compact Hausdorff space $Q$. If $f : Q \to X$ is continuous and if $\overline{co}(f(Q))$ is compact in $X$, then the integral $$ y = \int_Q f d \mu $$ exists, moreover $y \in \overline{co}(f(Q))$
Proof below:
Regard $X$ as a real vector space. Put $H = co(f(Q))$. We have to prove that there exists $y \in H$ such that $$ \Lambda y = \int_Q (\Lambda f) d \mu \;\; (2) $$ for every $\Lambda \in X^*$ Let $L = \left\{ \Lambda_1, ..., \Lambda_n \right\}$ be a finite subset of $X^*$. Let $E_L$ be the set of all $y \in \overline H$ that satisfies (2) for every $\Lambda \in L$. Each $E_L$ is closed (by continuity of $\Lambda$ and therefore compact, since $\overline{H}$ is compact.
Now there's the first bit I don't get it, despite I've tried to use the relevant theorem I can't manage anyway.
If no $E_L$ is empty, the collection of all $E_L$ has the finite intersection property. The intersection of all $E_L$ is therefore not empty, and any $y$ in it satisfies (2) for every $\Lambda \in X^*$.
Can you please clarify those lines (like expound the details)?
The other bit I don't get is how theorem 3.20 and theorem 3.4 are used to prove the following
Regard $L = (\Lambda_1,\ldots,\Lambda_n)$ as a mapping from $X$ into $\mathbb{R}^n$, and put $K = L(f(Q))$. Define $$ m_i = \int_Q (\Lambda_i f) d\mu \;\; (1 \leq i \leq n) $$ We claim that the point $m = (m_1,\ldots, m_n)$ lies in the convex hull of $K$. If $t = (t_1,\ldots,t_n)$ is not in this hull, then [by theorem 3.20 and (b) of theorem 3.4 and the known form of the linear functionals on $\mathbb{R}^n$] there are numbers $c_1,\ldots c_n$ such that $$ \sum_{i=1}^n c_i u_i < \sum_{i=1}^n c_i t_i $$ if $u = (u_1,\ldots,u_n) \in K$
How exactly the mentioned theorems are applied?
For the relevants theorems:
And
