# Radon-Nikodym property in Diestel & Uhl: a definition clarification

I posted the following question on MSE originally because, not being research-level, it seemed more appropriate for that site. However, there was no activity for it on MSE and I feel that it certainly can be answered on MO, so I am now posting it here.

A number of conditions that are equivalent to the statement that a Banach space $$X$$ has Radon-Nikodym property (RNP) are found on page 217 of Vector Measures" by Diestel & Uhl. One such condition is that: a Banach space $$X$$ has the RNP iff every function $$f:[0,1]\to X$$ that is of bounded variation is weakly differentiable off a fixed set of measure zero.

To me, the condition that $$f:[0,1]\to X$$ is weakly differentiable off a fixed set of measure zero" means that for a.e. $$t\in(0,1)$$, there exists a vector $$x_{t}\in X$$ such that $$\lim_{h\to 0}\frac{(\varphi\circ f)(t+h)-(\varphi\circ f)(t)}{h}=\varphi(x_{t})$$ for every $$\varphi\in X^{*}$$. Note that that this interpretation of weak differentiability at a point $$t\in(0,1)$$ matches the definition found in this paper at the bottom of page 2802 (the second page of the introduction).

First question: Is this interpretation of the condition that $$f:[0,1]\to X$$ is weakly differentiable off a fixed set of measure zero" correct? If not, what is the correct interpretation?

For example, one might reasonably also say that $$f:[0,1]\to X$$ is weakly differentiable if there exists a function $$g:[0,1]\to X$$ such that, for every smooth test function $$\varrho:[0,1]\to\mathbb{R}$$ with compact $$\text{supp}(\varrho)\subset(0,1)$$, it follows that $$\int_{0}^{1}\varrho'(t)f(t)dt=-\int_{0}^{1}\varrho(t)g(t)dt$$. This is, in turn, equivalent (see here, Prop. 6.36) to the dual condition that $$\int_{0}^{1}\varrho'(t)(\varphi\circ f)(t)dt=-\int_{0}^{1}\varrho(t)(\varphi\circ g)(t)dt$$ for every $$\varphi\in X^{*}$$. Clearly, if $$g:[0,1]\to X$$ is defined for a.e. $$t\in[0,1]$$ by $$g(t)=x_{t}$$ (where $$x_{t}\in X$$ is as stated above the first question) then the dual condition holds at least formally.

Second question: Is the dual condition for weak differentiability equivalent to the interpretation of weak differentiability above the first question?

2. Even in case $$X=\mathbb R$$ (which certainly has the RNP) the condition is not satisfied if interpreted in the second way: Cantor's stairway function $$f$$ is monotone and thus of bounded variation, and for $$f$$ the only candidate for $$g$$ in the second interpretation is the $$0$$-function.
• Thanks for your answer. I had the same thought about the phrase off a fixed set of measure zero" and Cantor's stairway function is a great example. Apr 27, 2022 at 14:20