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I intend to study Fourier Analysis, but lack the necessary background in measure theory. I have studied basic analysis and know uniform convergence etc. Are there some resources that contain the required measure theory and are brief?

Also, do I need to know any functional analysis as well?

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    Fourier analysis is very heavily based on measure theory and functional analysis. You can get some of the basic ideas out of it without either, but the proper treatment requires decent knowledge of both. – 2017-02-17
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    A typical example is Folland's book, *Real Analysis*. Chapters 1-3 are measure theory and chapters 4-7 are functional analysis (with some overlap). You have to have digested practically all of it before you are ready to read chapter 8, which is on Fourier analysis. – 2017-02-17
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    The 2nd part of B. Z. Vulikh's "Brief Course in the Theory of Functions of a Real Variable" will give you an excellent, clean, well-paced exposition of measure theory. This is a monograph. For exercises, I would look into the book by Kolmogorov and Fomin. This book is also not bad, but not nearly as clean and complete as Vulikh's. – 2017-02-17
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    @NateEldredge, Folland's book gives a competent expositions, but, IMHO, crams too much material into too little space to be a good first exposition. I would use Folland *after* getting the basics of measure theory. – 2017-02-17

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