Let $d\mu$ be a centered Gaussian measure on $\mathbb{R}^n$ with the covariance matrix $\sigma$ and $X : \mathbb{R}^n \to \mathbb{R}^n$ be any measurable mapping. Or we can simply regard $X$ as a random vector.
Now, define the pushforward $d[X_* \mu]$ of $\mu$ by $X$ as \begin{equation} [X_* \mu](E):=\mu(X^{-1}(E)) \end{equation} for any Borel set $E \subset \mathbb{R}^n$.
Also, write $X=(X_1, \cdots, X_n)$ where each $X_i : \mathbb{R}^n \to \mathbb{R}$ is a random variable for $i=1,\cdots,n$.
Then, is it true that \begin{equation} \int_{\mathbb{R}^n} X_i(x) X_j(x) \text{ }d[X_* \mu](x)= \sigma_{ij} \end{equation} for $i,j=1,\cdots,n$, where $\sigma_{ij}$ is the $(ij)$-entry of the covariance matrix $\sigma$.
I think this formula is a bit of a tautology but cannot verify it rigorously..
Could anyone help me?