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.

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. $\endgroup$ – Christian Chapman Nov 1 '17 at 3:26