I am requesting some references to learn appropriate measure theory for PDEs. Specifically, I would like to learn all the measure theory necessary to understand well-posedness of PDEs with measure valued data and things like that (I prefer weak/Sobolev spaces). I really would like something succinct without too much extra detail, so a book called "Measure Theory" would probably be a bad suggestion. Maybe a set of lecture notes designed for PDE people would be good?

I would like to avoid all the minute details and lemmas that I will never remember if possible. Thanks.

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    $\begingroup$ It's probably more of a light supplement but, try the appendix of Evan's PDE book. $\endgroup$ – Alex R. Jul 22 '13 at 16:39
  • $\begingroup$ Bogachev's Measure Theory seems what you need. $\endgroup$ – Tomek Kania Jul 22 '13 at 16:43
  • $\begingroup$ Related: mathoverflow.net/questions/136798 $\endgroup$ – Christian Clason Jul 22 '13 at 22:09
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    $\begingroup$ @TomekKania: could you please elaborate. The book you mention is a two volume treatise totaling more than thousand pages. At a glance this strikes me as the opposite what OP is looking for, but perhaps I am missing something. $\endgroup$ – user9072 Jul 22 '13 at 22:43
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    $\begingroup$ I would also ask the OP to clarify a bit. The phrase "measure valued data" and "weak/Sobolev spaces" seem a little bit odd to me. What exactly is it meant by "measure valued data"? Do you mean you want to study low regularity problems where the data is a measure on your space (as opposed to, say, a Lebesgue measurable function)? $\endgroup$ – Willie Wong Jul 23 '13 at 9:46

This supposed to be a comment but I can't comment yet. Evans also has a book "measure theory and fine properties of functions" coauthored with Gariepy if I'm not mistaken. It is more about geometric measure theory, but has a lot of interesting stuff about Sobolev spaces among other things.

  • $\begingroup$ Thank you. I was hoping for something lighter than this book but maybe I ask for too much. $\endgroup$ – Chris_A Jul 24 '13 at 7:31

Since your question is formulated somewhat provocatively, let me answer in a similar spirit: The only measure theory you need to know is that the space of Radon measures is the topological dual of the space of continuous functions. And since you're not interested in details and lemmas, you're already done.

Put less facetiously, what you need from measure theory is mainly the Riesz representation theorem in sufficiently general form. This is already treated in textbooks on abstract analysis (as part of integration theory), and you can avoid books on (modern) abstract measure theory (which evolved, in part, as a foundation for probability theory). A list of suitable texts can be found in Measure theory treatment geared toward the Riesz representation theorem, to which I would add Conway's recent Course in Abstract Analysis, AMS 2012.

(If you're interested in even "cooler" right-hand sides, you really need to make your question more specific.)


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