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.