Want to study Graduate Measure Theory with heavy Emphasis on Topology and/or Geometry.

Question

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.