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I have taken advanced courses both in measure theory and also in functional analysis (Banach and Hilbert spaces, spectral theory of bounded and unbounded operators, etc.)

The connections between the two arises in several theorems:

  1. Riesz theorem showing that under some conditions a continuous functional can be represented as integral with respect to some measure.

  2. Spectral measure and functional calculus for the bounded/unbounded self-adjoint operators.

I have also seen some other results that state that the dual of specific Banach spaces are the same than those of finitely additive measures.

In spite of having advanced course, the connection between measure theory and functional analysis is still really mysterious to me.

I would like to learn more about the connection between the two subjects in a more systematic fashion. I have already seen several related books but the connection is discussed only superficially.

I was wondering if anyone has a suggestion for a rigorous book that focuses specifically on the connection between measure theory and functional analysis.

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    $\begingroup$ Can you be more specific about what you want? My impression is that practically every graduate-level analysis book discusses these topics and the connections between them. What books have you read, and what was it about them that seemed "superficial" to you? What were you looking for that you didn't find? $\endgroup$ – Nate Eldredge Oct 31 '17 at 19:51
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    $\begingroup$ In my opinion Riesz-type theorems shouldn't be viewed as a "connection between measure theory and funcitonal analysis". You should just get used to measures as standard entities, as standard as functions, and the two are in duality to each other via integration. Riesz theorem only ascertains that some specific type of functions corresponds to some specific type of measures. $\endgroup$ – erz Oct 31 '17 at 21:38
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    $\begingroup$ Is Analysis Now by Pedersen one of the books you consider to have a superficial treatment? If it is to the appropriate level you might specifically be interested in the last chapter: $${}$$ > This chapter has two functions: Throughout the book it has served as an Appendix, to which the reader was referred for definitions, arguments and results about measures and integrals. It will now serve as a functional analyst's dream of the ideal short course in measure theory. $${}$$ It's placed after a rigorous development of spectra by Gelfand transform and an overview of unbounded operators. $\endgroup$ – enthdegree Nov 1 '17 at 3:26
  • $\begingroup$ Thanks all. @Nate: what I meant by "superficial" was not about the content or beauty of the results. What I meany was that the connection between them seems (at least with my current knowledge) as disjointed pieces of a bigger picture and I am interested to understand the connection deeper. Let's consider the functional analysis itself. Before FA, we had disjointed pieces of a puzzle (functions, sequences, ....) which where treated differently but after FA, now we can see the underlying connection clearly. I am looking for such a connection between disjoint occurences of measure theory in FA. $\endgroup$ – Saeid Haghighatshoar Nov 2 '17 at 13:57
  • $\begingroup$ "Functional analysis" is a very broad term. One of its meanings is "the study of spaces of functions". If you accept that, a key connection between functional analysis and measure theory is the duality between spaces of functions and spaces of measures. A big-picture paper you may like is "A survey of Baire measures and strict topologies" by R.F. Wheeler, Expositiones Mathematicae 2 (1983) 97--190. It is not a book, but the length is book-like. $\endgroup$ – user95282 Nov 6 '17 at 17:55
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Very popular are Walter Rudin's books, 1. Functional analysis and 2. Real and Complex Analysis. They cover substantially more than Kolmogorov-Fomin, and from a more modern point of view than Riesz-Nagy.

Another excellent choice is MR0662563
Malliavin, P. Intégration et probabilités. Analyse de Fourier et analyse spectrale, Masson, Paris, 1982. There is an English translation. It has much more measure theory but less complex analysis.

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You can take a look at the book by A.N.Kolmogorov and S.V.Fomin Elements of the theory of functions and functional analysis. They discuss measure theory and its connection with functional analysis (and they prove the Riesz theorem).

There is also a good book by F.Riesz and B. Sz.-Nagy Functional analysis (the first author is the very same F.Riesz who proved the theorem you are talking about).

About spectral theorem you can also read in A.Ya.Helemskii's Lectures and Exercises on Functional Analysis.

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You have to go fairly far with measure theory and functional analysis in order to use one to understand the other better. There are some intersections such as $L_p$-spaces and integral representations. Books that will teach you about happy marriages of measure theory and functional analysis are "Lectures on Choquet's theorem" by Robert Phelps and the first few chapters of "Optimal Transport: Old and New" by Cedric Villani. There is also the rich topic of vector measures; the book "Vector Measures" by Diestel and Uhl is surprisingly readable.

On a deeper level, measure theoretic results rely heavily on order structure and this is where measure theory and functional analysis have deep connections. One can, for example, obtain the Hahn decomposition from a general result valid in all vector lattices, but vector lattices are not a topic usually taught in introductory functional analysis courses. A not so gentle read on the connection between vector lattices and measure theory is "Topological Riesz Spaces and Measure Theory" by David Fremlin.

As an aside: In the heyday of Bourbaki, it used to be popular to reduce measure theory to integration theory and integration theory to a study of dual spaces via the Riesz representation theorem. This works reasonably well when doing topological measure theory on locally compact, but works in general only via some clumsy constructions using sophisticated compactification arguments. In probability theory, the most cheerful importer of measure theory, one regularly has to deal with measures on function spaces that are not locally compact, so the approach via the Riesz representation theorem is of rather limited usefulness.

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  • $\begingroup$ Thanks a lot for the detailed response. Interestingly, I was looking for a suitable book in the library and I found "Vector Measures" by by Diestel and Uhl which goes deeper in this direction. I am now reading this book. I also checked "Topological Riesz Spaces and Measure Theory" by David Fremlin. It seems that the author addresses exactly the same connection that I am interested in although for the specific case of Riesz spaces. I will definitely check this book. Thanks a lot. $\endgroup$ – Saeid Haghighatshoar Nov 2 '17 at 14:15
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There is a new book in five volumes by Barry Simon: "A Comprehensive Course in Analysis". The first volume, in particular, and maybe also the last one on spectral theory would be ideal references for the OP's topics. I looked at parts of this new series and I think it is really good. I like in particular Simon's "Kvetches" which I can paraphrase as stay away from non-Borel measurable functions or don't go too far from metrizable spaces. The chapter on wavelets is amazing (gives a construction of Meyer and Daubechies wavelets in less than thirty pages).

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  • $\begingroup$ Thank you very much for the suggestion. This series by Barry Simon is also my favorite ones; I have read mainly the complex analysis part. I will check the parts you mentioned definitely. Thanks a lot. $\endgroup$ – Saeid Haghighatshoar Nov 2 '17 at 14:18

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