TITLE: Good book for measure theory and functional analysis
QUESTION [15 upvotes]: 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.

REPLY [6 votes]: 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.