From the book Infinite ergodic theory of numbers, Kesseböhmer, proposition 2.4.27b on page 106. I'm sorry but I don't have a link.
They say it follows directly from Maharam's theorem. They probably mean that there is a sweep-out set $Y$ and measure preservingness is satisfied by part a (from the same proposition 2.4.27). But then what about the $\sigma$-finiteness? I don't think that is so trivial and 'directly from Maharam'.
From Kac's lemma it is easy, from which it follows that $m(X)<\infty$. But the proof of Kac's lemma is not trivial, so then we still don't have a direct consequence of Maharam.
Maybe they mean the $\sigma$-finiteness is already in their definition of 'system'? In the beginning of section 2.1 they only say that about 'dynamical systems'.
I hope someone can give an answer. Thanks in advance.