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I did one course in Measure Theory and want to study it again. But this time I want to do this in a way that emphasizes Measure Theoretic structure on Geometric or Topological Spaces. I don't know, if it is at all possible. But I have heard there are some Differential forms and Measure theoretic relations on manifolds. So, my question is,-

  1. Do you know a graduate level literature (book,notes,lectures anything) on Measure Theory that you think is topologically or Geometrically heavy (in your judgement) ? Showing an interplay between Measure theory and topology/geometry is desired but any other reasons that you have is good too.

  2. More specifically, Is there anything for measure theory on manifolds?

  3. Is reading Geometric Measure Theory a good idea for this?

I asked this on Reddit too. But I need more opinions.Thank You.

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  • $\begingroup$ I've heard many things about Federer's book on Geometric Measure Theory – not all of them good... It might be of interest to you, though. $\endgroup$ – Alekos Robotis Jul 13 at 4:44
  • $\begingroup$ I'm not sure GMT is what you are looking for (unless you are interested in things like minimal submanifolds). Maybe you are looking for smooth ergodic theory or something of that nature? See, e.g., this book or this PDF. $\endgroup$ – user10354138 Jul 13 at 7:20
  • $\begingroup$ Maybe "Geometric Measure Theory" by F. Morgan. $\endgroup$ – d.k.o. Jul 13 at 8:00
  • $\begingroup$ When you say "emphasizes Measure Theoretic approach over Geometric or Topological Spaces" I take it you mean something like "measure theory on geometric spaces" rather than "measure theory instead of geometric/topological spaces"? $\endgroup$ – T_M Jul 13 at 8:07
  • $\begingroup$ @T_M yes, I mean Measure Theory on Geometric Spaces. $\endgroup$ – Sagnik Biswas Jul 13 at 8:18

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