I have recently been reading Werner Ballmann's Introduction to Geometry and Topology, and when I get to section 7 of chapter 3 (where the author wants to define the oriented integral), he says "Let $A \subset M$ be Lebesgue-measurable...", where $M$ is a smooth manifold. I have studied some basic measure theory, mainly from Rudin's Real and Complex Analysis and Cohn's Measure Theory, but I have not understood how to define Lebesgue measure on a general manifold, or, for that matter, what it means for a subset of a manifold to be Lebesgue-measurable. Therefore, I would like to know how Werner Ballmann defines Lebesgue measure in this context, and I do not seem to have found any other sources that define Lebesgue measure on manifolds. Finally, I would like to share my own thoughts. According to the definition of zero measure sets on manifolds, I guess that what Werner Ballmann means by "a subset $A$ of a manifold $M$ is Lebesgue measurable" is that for any chart $(U,x)$ with $U \supset A$ in the maximal atlas, $x(U)$ is Lebesgue measurable?
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$x(A\cap U)$ needs to be measurable. I have written several answers about measures on manifolds; see Surface measure, and Lebesgue spaces on Riemannian manifolds. There are many other answers (some which you can find linked to these); that should address your questions. – peek-a-boo Feb 22 '24 at 08:45
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And here is a good-to-know fact about Radon-Nikodym derivatives of scalar density-induced measures on manifolds, measure-zero sets, and here is a similar discussion on measure-zero sets but from the Riesz point of view. – peek-a-boo Feb 22 '24 at 08:52