I've been attempting to learn and understand Lebesgue integration over the past few days, and have a question about an observation I have made about the Lebesgue integral with my currently limited understanding of it.
I take the limit of the Riemann sum, $\displaystyle\lim_{n \to \infty} \sum_{i=0}^{n} \mu(F^{-1}([y_{i+1}, y_i])) \cdot \Delta{y}$, the sum of the measures of discrete sections the inverse images of our total area $A$ rendered as a set of points. This roughly follows the format of a normal Riemann integral, merely passed through a different function which removes some limitations from what can be integrated, such as limitations concerning continuity.
I would then propose that any aspects of the Fundamental Theorem of Calculus that are not dependant on the generalizations introduced by passing our original function through a Lebesgue measure before integrating across its range would still hold, as fundamentally the Lebesgue integral is a generalization of the Riemann integral for a greater set of measurable functions. Am I missing anything fundamental about this statement which either alters it or proves it false?
More practically speaking, is there a Fundamental Theory of Calculus for Lebesgue Integrable functions such as indicator functions, which does not depend on continuity? If so, how is it different or similar to the notion of a Fundamental Theory of Calculus without dependence or conclusions based on continuity (assuming that statement isn't complete nonsense, which I think it actually could be).
There is a Fundamental theorem of calculus for Lebesgue integrable functions. More precisely, if $g$ is a Lesbesgue integrable function, then the function $f(x)=\mu((-\infty,x])$ associated to the measure $$\mu(A)=\int_Ag \ dx\;,$$ with $dx$ the Lebesgue measure, is absolutely continuous. You can read more about absolute continuity from wikipedia https://en.wikipedia.org/wiki/Absolute_continuity. Moreover, any absolutely continuous function has an associated Lebesgue integrable derivative (called the Radon-Nikodym derivative). In this weaker sense, the fundamental theorem of calculus holds.