Nonnegative Polynomials Imply Nonnegative Probability Measure Product?

Question

Let $f$ and $g$ be polynomials over $\mathbb R^n$. Let $X \subset \mathbb R^n$ be a compact, convex set (a polyhedron, specifically, if that makes a difference).

Is the following true?

Claim: If $f(\mathbf x) g(\mathbf x) \geq 0$ for all $\mathbf x \in X$, then

\begin{equation} \left ( \int_X f dP \right ) \left (\int_X g dP \right ) \geq 0 \end{equation} for all probability measures $P$ over the set $X$.

Answer 1

Try $f(x) = (x+2)x(x-1)$, $g(x) = -f(-x)$ on $X=[-1,1]$.

Then $f(x)g(x) \ge 0$, but with $P= {1 \over 2} m$ (that is, half the Lebesgue measure) we have $\int f dP = {1 \over 3}, \int g dP= - {1 \over 3}$.