Let $g:\mathbb{R}^n\rightarrow \mathbb{R}_{++}$ which is twice differentiable and non-constant. I want to show that if $$\int g(x)\,\text{d}(\Phi(x;0,\Sigma)-\Phi(x;0,\tilde\Sigma))=0, $$ then $$\Phi(x;0,\Sigma)=\Phi(x;0,\tilde\Sigma).$$ $\Phi(x;0,\Sigma)$ is a multivariate normal distribution with mean zero and variance $\Sigma$. $x$ is a random vector (number if $n=1$).
I guess this is true statement, I am not sure how I should show it. If there is a counter example, I wish to see that.