For a Banach space $E$ and a probability space $(X,𝒜,\lambda )$, a new proof is given that a measure $\mu :𝒜\to E$, with $\mu \ll \lambda$, has RN derivative with respect to $\lambda$ iff there is a compact or a weakly compact $C\subset E$ such that ${\left|\mu \right|}_C:𝒜\to [0,\infty ]$ is a finite valued countably additive measure. Here we define ${\left|\mu \right|}_C\left(A\right)=sup\{{\sum }_k|\langle \mu \left(A_k\right),f_k\rangle \left|\right\}$ where $\left\{A_k\right\}$ is a finite disjoint collection of elements from $𝒜$, each contained in $A$, and $\left\{f_k\right\}\subset E^{\text{'}}$ satisfies ${sup}_k\left|f_k\left(C\right)\right|\le 1$. Then the result is extended to the case when $E$ is a Frechet space.

We present some consequences of a deep result of J. Lindenstrauss and D. Preiss on $\Gamma$-almost everywhere Fréchet differentiability of Lipschitz functions on $c_0$ (and similar Banach spaces). For example, in these spaces, every continuous real function is Fréchet differentiable at $\Gamma$-almost every $x$ at which it is Gâteaux differentiable. Another interesting consequences say that both cone-monotone functions and continuous quasiconvex functions on these spaces are $\Gamma$-almost everywhere Fréchet differentiable....