Given two absolutely continuous probability measures $\mu,\sigma \in \mathcal P_2(\mathbb R^n)$ and two maps $T_1, T_2$ such that $$(T_1 \circ T_2)_\#\sigma =\mu$$ where $(\cdot)_{\#}$ denotes the pushforward operator. I saw that it is a general property that
$$(T_1 \circ T_2)_\#\sigma={T_1}_{\#}({T_2}_{\#}\sigma)$$
How can one make sense of this or see this?
If I'm not wrong, for a probability $\mu$ and a map $T$, one defines $$T_\# \mu(A) := \mu(T^{-1}(A)),$$ for a measurable set $A$, right?
Then, in your notations: $$\begin{align*} (T_1 \circ T_2)_\#\sigma(A) & = \sigma((T_1\circ T_2)^{-1}(A)) \\ & = \sigma(T_2^{-1}(T_1^{-1}(A))) \\ & = (T_{2\#}\sigma)(T_1^{-1}(A)) \\ & = T_{1\#}(T_{2\#}\sigma)(A). \end{align*}$$ That is, $$(T_1 \circ T_2)_\#\sigma = T_{1\#}(T_{2\#}\sigma).$$