I'm looking for sources on non-commutative geometry and integration theory. I wonder if this theory might replace the standard theorey in the long run, as it seems to be more general. What are possible applications of this theory to physics? Connes has a lot to say about the Standard Model, but I don't really understand what that theory gains from Conne's point of view. I haven't been able to get an overview of what learning non-Commutative geometry exactly entails and what it is used for.
I tried reading the book by Connes, but got stuck very early, at the part where he starts going on about leaves and foliations. I have some background in Hilbert space theory (Unbounded operators, spectral theorems, Schatten classes) and also very basic manifold theory (Vector bundles, basic statements about de Rham cohomology...). I was planing to learn the foliations stuff anyway because of the connections to PDE, but haven't been able to find sources I like so far.
I tried looking for other sources on non-Commutative measure theory to get into the subject a bit, but found only very specific and technical accounts of certain aspects of the theory. Does anyone know good sources to start leaning this?