Understanding a lemma from Bauer's book

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This lemma precedes the proof of the Radon-Nikodym Theorem in Bauer's book Measure and Integration Theory.

At the beginning of the proof he writes "we may obviously suppose that $\varrho(\Omega)\geq0$", but I can't find anywhere in the proof where this assumption is needed. Am I missing something?

Thanks a lot for your help.

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