Let $A$ be an open measurable set and $A_0$ is a subset of $A$ which has measure zero.
Let $g$ and $f$ be almost everywhere differentiable function defined on $A$.
What is the impact of "almost everywhere"? What do I need to watch out for?
For example can we still use integration by parts and write
$$\int_A f(x) g^{\prime}(x) d x=f(x) g(x)- \int_A f^{\prime}(x) g(x) d x$$
Or for example, since $f$ is a.e. differentiable we know that it is also a.e. continuous, therefore do we still have the convergence of the Riemann sum to the Riemann integral, i.e.
$$\int_a^b f(x) d x= \lim_{n \rightarrow \infty} \sum_{ i = 1 }^n f( a+i \Delta x) \Delta x$$
where $a,b \in A$ and can be such that $a, b \in A_0$