Let $p(x)$ be a probability density function on the unbounded set $X \subseteq \mathbb{R}^n$, so that $\int_X p(x) dx = 1$.

Let $F: X \rightarrow \mathbb{R}_{\geq 0}$ a measurable but non-integrable function, i.e.

$$ \int_X F(x) p(x) dx = \infty $$

I'm wondering if the following proposition is true:

$ \forall \text{ such } F(\cdot) \ \ \exists $ a strictly-increasing, concave function $ f: R_{\geq 0} \rightarrow R_{\geq 0} $, with $f(0) = 0$, $\lim_{y \rightarrow \infty} f(y) = + \infty$ such that:

$$ \int_X f(F(x)) p(x) dx < \infty $$