Take any function $f:\mathbb R^n\to\mathbb R$. The convex hull of $f$ can be defined as
$$ f^{**} := \sup\{g \text{ is convex }: g(x) \leq f(x)\;\forall x\}. $$
One way to find this is to take the limit of $p\to\infty$ of the following problem $$ f^{**}(x) = \min_{\theta}\left\{\sum_{i=1}^p \theta_i f(z_i) : \sum_{i=1}^p \theta_i z_i = x, \;\theta_i\geq 0,\; \sum_{i=1}^p \theta_i = 1\right\}. $$
I'd like to write this statement in integral form, but I'm struggling, and for some reason I can't seem to find a reference.
Best so far from comments: $$ f^{**}(x) = \min_\mu\left\{\int_{\mathrm{dom} f} f(\mu) d \mu : \int_{\mathrm{dom} f} \mu d \mu = x\right\}. $$