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

Question

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.

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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?

Answer 1

Diestel & Uhl can only mean the first interpretation you gave for two reasons:

1. For the second interpretation, the term “off a fixed set of measure zero” makes no sense.
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.

The latter shows also that the answer to your second question is negative.