Good book for measure theory and functional analysis 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:


*

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

*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.   
 A: 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.
Added. Let me also mention E. Stein and R. Sharkachi, 4 volumes which cover all standard graduate analysis curriculum (Fourier, Complex, Real and Functional analysis). I like and recommend this.
A: 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.
A: 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.
A: 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).
