# Lebesgue's Majorized Convergence Theorem

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.

-
Pretty sure this is the dominated convergence theorem. – Andrés E. Caicedo Jan 9 '11 at 23:59
right different ways to translate the same thing into English – Gerald Edgar Jan 10 '11 at 12:12

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 $G' =\sum_{n=1}^{\infty} |f_{k}(x)|$.