Last semester, I was in a complex analysis class where one of the assignments was to give a presentation on some subject in complex analysis. I chose to lecture on Fourier analysis, with an emphasis on music. Steven Krantz attended, said he liked the lecture, and then pitched me a book sale. Interested in learning more, I finally ordered it last week. So now I have this book:
and it seems to be pretty far above my head. If someone could recommend to me how I should approach this book, I would be very grateful.
In the preface, he talks about the history of Fourier analysis, which I can understand up until he mentions singular integrals, Hilbert transforms, Hardy spaces and so on. I've only taken a few calculus classes so I have no idea what any of that means. My hopes are I'll get it by the end of the book. The first actual chapter isn't reassuring so far:
This is where I start to get lost. I get the notion of open sets and subsets, and "pairwise disjoint" makes sense to me, but that big cap operation with all those is a foreign thing. The book the goes on to describe m(U) as what I think is supposed to be the sum of the lengths of all the . The next few pages are full of operations like and it's very intimidating.We take it for granted that the reader of this book has some acquaintance with elementary real analysis. ... However, the reader may be less acquainted with measure theory. The review that we shall now provide will give even the complete neophyte an intuitive understanding of the concept of measure, so that the remainder of the book may be appreciated.
Now let us look at these matters from another point of view. Let be any subset. If is any open set, then of course we may write
where each is an open interval and the are pairwise disjoint.
My plan right now is to go to the library and start reading textbooks on real analysis and set theory (I haven't had the opportunity to take classes on either of these yet), and perhaps on measure theory if it helps. Is this doable? Is it a lost cause?
(edit: when I was uploading the LaTeX pictures to Tinypic they gave me captchas with Greek letters. Come on.)