# Factorization of Square-integrable random-variables and Generalized Inverses

Suppose that $$X,Y,Z \in L^2(\Omega,\mathcal{F},\mathbb{P};\mathbb{R}^d)$$, are $$d$$-dimensional random-vectors and there exists functions $$f,g\in L^2(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d),Law(X);\mathbb{R}^d)$$ satisfying $$f(X)=Y \qquad g(X)=Z.$$

If $$f$$ is invertible, then the function $$h\triangleq g\circ f^{-1}$$ implies that $$Z=h(Y).$$

However, if $$f$$ is not invertible, when does there exists a function $$h$$ satisfying $$h\circ f = g$$ except possibly on a set of $$Law(X)$$-measure $$0$$? Hence $$h(Y)=Z .$$

• The law of $X$ is a measure on the Borel sets of $\Bbb R^d$. Something does not add up here. – Mars Plastic Mar 12 at 13:29
• Thanks, made it clear now :) – AIM_BLB Mar 12 at 13:34
• The space $L^2(\Omega,\mathcal{F},Law(X);\mathbb{R}^d)$ still doesn't make sense. ;-) – Mars Plastic Mar 12 at 13:37
• Should be good now, thanks for noticing that; I fixed it to the Borel measure space on $d$-dimensional Euclidean space. :) – AIM_BLB Mar 12 at 13:47