A doubly-continuous knapsack problem is this sort of maximization problem understood? 
Take a function $f:\mathbb{R}_+^n\to \mathbb{R}_+$ increasing in each dimension, differentiable, bounded above by some constant $C$. (possibly not concave). Fix $v\in\mathbb{R}^n_+$.
\begin{align} 
    \text{maximize }\ & \int f(x)\mu(dx) \\
    \text{over }\ & \mu \text{, a distribution on } \mathbb{R}_+^n \\
    \text{subject to }\ & E_\mu[x]=v.
\end{align}
 A: Here is a finite point mass result that uses Caratheodory's theorem.

Define 
$$A = \{(x, f(x)) \in \mathbb{R}^{n+1} : (x_1, ..., x_n) \geq 0\}$$
Let $X$ be any nonnegative random vector and assume $E[X]$ and $E[f(X)]$ are both finite. Then $(X,f(X)) \in A$ always, and so,
$$ E[(X, f(X))] \in Conv(A) $$
where $Conv(A)$ is the convex hull of the set $A$. 
Fix $v=(v_1, ..., v_n)$ as a vector with positive entries. We want to maximize $E[f(X)]$ subject to $E[X]=v$. This is equivalent to
\begin{align}
\mbox{Maximize:} \quad & y \quad (Eq. 1) \\
\mbox{Subject to:} \quad & x=v \quad (Eq. 2)\\
& (x,y) \in Conv(A) \quad (Eq. 3) 
\end{align} 
For simplicity assume a solution $(x^*,y^*)$ exists.  Since $(x^*, y^*) \in Conv(A)$, and $A$ has $n+1$ dimensions, Caratheodory's theorem ensures $(x^*,y^*)$ can be written as the convex combination of at most $n+2$ points in $A$.  However, since $(x^*, y^*)$ is on the boundary of $Conv(A)$, a simple extension to Caratheodory ensures $(x^*,y^*)$ can be written as a convex combination of at most $n+1$ points in $A$, so there are probabilities $p_1, ..., p_{n+1}$ that sum to 1, and nonnegative vectors $x_1, ..., x_{n+1} \in \mathbb{R}^{n}$, such that: 
$$(x^*, y^*) = \sum_{i=1}^{n+1} p_i(x_i, f(x_i)) $$
That is, we define an optimal random variable $X^*$ to take the value $x_i$ with probability $p_i$, for $i \in \{1, ..., n+1\}$, and this solves the problem. $\Box$
