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.In fact, I believe this paper obtains a slightly stronger result showing that one has divergent examples in the space $L \log^{1-\epsilon} L$.
