Lebesgue's Majorized Convergence Theorem

Question

Can anyone point me to an explanation and a proof of this theorem?

For reference, it is mentioned in Kolmogorov's almost everywhere divergent function in $L$ as given in Zygmund, volume I. In the third edition it can be found at the top of page 306. A google search leads to several references to this theorem (the Marjorized Convergence Theorem, that is, not Kolmogorov's) in research papers but nothing that would lead a relative neophyte to an understanding.

I'm happy to be advised to go and buy a certain book. I'm reading Alan Weir's 'Lebesgue Integration and Measure' at the moment, but I suspect I need a more extensive treatment for the future.

I hope I won't have egg on my face and someone tell me "It's just the Dominated Convergence Theorem, silly!" I feel I should be able to read Zygmund and ascertain this myself but the reasoning is mixed with results relating to Fourier series and I cannot unravel it.

Many thanks in advance.

Answer 1

I also believe that this just is the dominated convergence theorem. The relevant line on the top of page 306 seems to be:

"The majorized convergence implies that S[f] is obtained by adding formally the Fourier series of the individual terms on the right of (3.4), that is by writing out in full the successive polynomials ...."

It seems that all he really is using here is the linearity of integrals/Fourier series, that is $\hat{G}(n)=\sum_{n=1}^{\infty }\hat{f_k}(n)$ where $G = \sum_{n=1}^{\infty} f_k (x)$ and $G' \in L$ where $G' =\sum_{n=1}^{\infty} |f_{k}(x)|$.

If you are trying to understand Kolmogorov's example of an integrable function with an almost everywhere divergent Fourier series, there are two alternate expositions that you may find helpful. Grafakos' Classical Fourier Analysis contains a complete proof of Kolmogorov's example (and is very thorough in providing details). A somewhat different but more conceptual "existence proof" is contained in E. Stein's paper On limits of sequences of operators. Ann. Math. 74(I), 140-171.