Differentiability of convex functions and Rybakov’s theorem

Abstract:

Rybakov proved that if $\mu :\Sigma \to X$ is a countably additive Banach space valued measure on a $\sigma$-algebra $\Sigma$, then there is an element ${x^ * } \in {X^ * }$ so that $\mu \ll {x^ * }\mu$. In this note, we show that Rybakov’s theorem follows essentially from a classical result of Mazur on the Gateaux differentiability of convex functions.