I'd be grateful for a reference for the following result, which I believe to be true, and 
should be well-known.


Let the continuous functions $f\_0,f\_1,\cdots,f\_n: [0,1]\rightarrow [0,\infty)$ be given
and consider the problem of maximizing the integral
$$\int\_0^1 f\_0(x)d\mu(x)$$
over all *positive* Borel measures $\mu$ on [0,1], which satisfy the constraints
$$\int\_0^1 f\_k(x)d\mu(x)=1,\;\;1\leq k\leq n.$$


Then, if a solution exists, the maximum is attained by a linear combination of (at most) $n$
shifted $\delta$-functions:
$$\mu=\sum\_{j=1}^n \alpha\_j\delta(x-x\_j),\;\;\;x\_j\in [0,1]$$



A simple infinite dimensional optimization problem
This is a particular case of the Generalized Moment Problem. 


The result you are looking for can be found in the [first chapter](http://www.worldscibooks.com/etextbook/p665/p665_chap01.pdf) of *Moments, Positive Polynomials and Their Applications* by Jean-Bernard Lasserre (Theorem 1.3).
The proof follows from a general result from measure theory.



> 
> **Theorem**. Let $f\_1, \dots , f\_m : X\to\mathbb R$ be Borel measurable on a measurable
>  space $X$ and let $\mu$ be a probability measure on $X$ such that $f\_i$ is
>  integrable with respect to $\mu$ for each $i = 1, \dots, m$. Then there exists a
>  probability measure $\nu$ with finite support on $X$, such that:
>  $$\int\_X f\_id\mu=\int\_Xf\_i d\nu,\quad i = 1,\dots,m.$$
>  Moreover, the support of $\nu$ may consist of at most $m+1$ points.
> 
> 
> 


