Let $([0,1]^n, \mathcal{B}, P)$ be a probability space where $\mathcal{B}$ is the Borel $\sigma$-algebra and $P$ is a probability measure.
Let $f:[0,1]^n\to \mathbb R^n$ be a measurable function (a random vector). The pushfoward measure $Q=f_{\#} P$ is defined by $Q(A) =P(f^{-1}(A))$ for all $A\in \mathcal{B}$.
Define a system of linear projections $f_1,\dots , f_n $ of $f$ onto each coordinate subspace $[0,1]$ where for each $i=1,\dots, n$, $f_i: [0, 1]\to \mathbb R$ is defined by
$$f_i(x_i) =\int_{[0,1]^{n-1}} f(x_i, x_{-i}) dP(x_{-i}|x_i)$$
where $x_{-i}= (x_1,\dots,x_{i-1},x_{i+1},\dots,x_n)$.
Define $\tilde f=(f_1,\dots,f_n)$. Then $\tilde f$ is also a random vector, where each $f_i$ depends only on $x_i$. Define the pushfoward measure $\tilde Q=\tilde f_{\#} P$.
What can we say about the relations between the pushfoward measures $Q$ and $\tilde Q$?
Is there any literature on such problems?
Thanks.