Our Mathematical analysis lecturer said the course (2 year BS MATH) will be more Lebesgue measurement and integration focused, rather that the traditional Riemann one. She said, even thought it is more difficult to understand, it finds more applications in different fields, like statistics...
We are studying manifolds from a Lebesgue M&I perspective. Most of the literature I know in Russian, explains manifolds from the Riemann integral perspective.
Could you recomend me literature in English or Russian explaining manifolds from a Lebesgue perspective?
PS.I understand this post might not belong to this community. If you know a website fitting my request let me know in the comments.
For an introduction to manifolds that uses the Lebesgue integral there is Vector Analysis by Janich (a great book in any case). But really, unless the course is analysis focused ($L^2$ space of functions on the manifold, differential operators, convergence of sequences of functions etc...), it shouldn't matter that much what kind of integral you use