When I started studying integration rigorously via the Riemann and Lebesgue integrals, one thing that struck me is that we loose completely the concept of indefinite integrals. Integrals of functions are only defined over a particular set. I was wondering if there is still a way of defining an indefinite Lebesgue integral.
2026-04-12 17:04:54.1776013494
Bumbble Comm
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Is there a notion of indefinite Lebesgue integral?
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Bumbble Comm
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The most useful thing connected to this is probably the Lebesgue differentiation theorem.
Such a thing as an indefinite integral is only defined on $\mathbb{R}$ anyway, so it's of limited use in the general theory of the Lebesgue integral.
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Really, there's not a notion of "indefinite Riemann integral". A Riemann integral is by definition a limit of Riemann sums. What you want is a Newton integral, which is an antiderivative.
Also, in math or physics I've found you don't really use Newton integrals much. Most of the time you're integrating over certain bounds, or if you're solving a differential equation you're integrating from "$0$" to position/time/etc, $x$/$t$/etc. which is basically what an indefinite integral is.