A Fundamental Theorem of Calculus for Lebesgue Integration, J. J. Koliha begins with the passage
Lebesgue proved a number of remarkable results on the relation between integration and differentiation. They can be briefly summarized as follows:
Theorem (Lebesgue): A function $f:[a, b] \rightarrow \mathbb{C}$ is Lebesgue integrable on $[a, b]$ if and only if there exists an absolutely continuous function $F_f:[a, b] \rightarrow \mathbb{C}$ such that $F_f^{\prime}=f$ almost everywhere on $[a, b]$. In this case $\int_a^b f(t) d t=F_f(b)-F_f(a)$.
How can we use this theorem to calculate a Lebesgue integral in the case when the integrand $f$ is Lebesgue integrable on $[a, b]$ and has a primitive $F$ on $[a, b]$ (that is, $F^{\prime}=f$ everywhere in $[a, b]$) ? The problem is, we cannot, unless we know that $F$ is absolutely continuous on $[a, b]$.
Throughout the rest of this post $f\in L^1([a,b])$, and I will use the notation $F_f$ to denote the (unique) absolutely continuous a.e.-primitive of $f$, i.e. the unique absolutely continuous function $F$ on $[a,b]$ which is a.e.-differentiable with a.e.-defined derivative $F'=_\text{a.e.}f$.
Koliha goes on to prove something very similar to the following result. I will use the formulation in this MSE post.
If $F$ is continuous on $[a,b]$ and differentiable except at (at most) a countable set of points $S$, with derivative (where defined) $F'=:f$ in $L^1([a,b])$, then the FTC holds.
The MSE post in fact shows that one can replace "differentiable" in the previous sentence with "differentiable from the right".
In other words given $f$ integrable on $[a,b]$, we know that for any countable set $S\subseteq [a,b]$, any continuous ($[a,b]\smallsetminus S$)-primitive of $f$ — i.e. any continuous $F$ on $[a,b]$ which is differentiable on $[a,b] \smallsetminus S$ with derivative $F'=f$ on $[a,b]\smallsetminus S$ — must automatically be exactly $F_f$ above (i.e. automatically upgraded to be absolutely continuous).
On the other hand, this is not true for general null sets $S$. Even taking $S$ to be the Cantor set $C$, we have the Cantor function which is a continuous ($[0,1]\smallsetminus C$)-primitive of the $0$ function, which is not absolutely continuous.
My question is:
Is there any characterization of the sets $S$ so that for any $f$ integrable on $[a,b]$, all continuous ($[a,b]\smallsetminus S$)-primitives $F$ of $f$ satisfy the FTC (i.e. are automatically upgraded to be the absolutely continuous function $F_f$)?
Is there an example of an uncountable such $S$?
I realized perhaps my question is equivalent to asking for a characterization of sets $S\subseteq [a,b]$ that can not support a singular continuous measure. Seems like MSE has some information Is the set of non-differentiable points for a singular continuous function nowhere dense?
