Factorization of smooth measures (Q1814632)

This paper gives some results on a decomposition of a differentiable (real-valued) measure \(\mu\) on a Banach space \(Z\) into a product of such measures on the mutually complementary subspaces \(X\) and \(Y\) of \(Z\). One of them is the following: let \(Y\) be one-dimensional with \(\ell\) the Lebesgue measure on it. Then \(\mu\) is differentiable in the direction of \(h\in Y\) if and only if there exist a measure \(\nu\) on \(X\) and a Borel function \(\varphi: Z\to\mathbb{R}\) such that 1) for each fixed \(x\in X\), the function \(y\mapsto\varphi(x,y)\) is absolutely continuous, 2) \(D_ h\varphi\) is \(\nu\times\ell\)-integrable, and 3) \(\mu=\varphi(\nu\times\ell)\). Here, for a \(\nu\times\ell\)-integrable function \(f: Z\to\mathbb{R}\), \(f(\nu\times\ell)\) is the measure on \(Z\) defined through \((f(\nu\times\ell))(A)=\int_ A fd(\nu\times\ell)\) for every Borel subset \(A\) of \(Z\).