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There's a function $g(x,y):\mathbb{R^+\times \mathbb{R}^+\rightarrow \mathbb{R}^+}$ with $g_1(x,y)>0$, $g_2(x,y)>0$, and $g_{12}(x,y)>0$. I conjecture that $$\int g(x,x)f(x)dx>\int\left(\int g(x,y)f(y)dy\right)f(x)dx$$ for an arbitrary probability density function $f$.

The motivation is from the fact that when $g(x,y)$ is separable, i.e. $g(x,y)=a(x)b(y)$, $E[a(x)b(x)]>E[a(x)]E[b(x)]$ for any probability distribution since $a(x)$ and $b(x)$ have positive correlation.

And I'm trying to prove this more general case. Is it true?

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