Theory of Functions of a Real Variable I Remarks on the Proof of the Hahn Decomposition Theorem

The proof I gave in class was perfectly correct ( except for one minor omission), but the last part was a little bit unclear, and I made it sound more complicated than it really should have been. So let me make the proof completely transparent. Recall that the situation under consideration was this: we had a signed measure ν on a measurable space (X,M). Knowing that such a measure cannot take both values +∞ and −∞, we were assuming WLOG that ν does not take the value +∞. Our goal was to find a decomposition of X as the union P ∪ N , where P ∈ M and N = X\P , such that P is ν-positive and N is ν-negative. (Recall that if E ∈M, then (i) E is ν-positive if ν(F ) ≥ 0 for every F ∈M such that F ⊂ E; (ii) E is ν-negative if ν(F ) ≤ 0 for every F ∈M such that F ⊂ E; (iii) E is ν-null if ν(F ) = 0 for every F ∈M such that F ⊂ E.) And here is the proof. (I will very sketchy in repeating what was done clearly and correctly in class, until I get to the part that needs further clarification; and from then all I will try to be completely clear and give all the details. And in addition I will take care of the omission mentioned above.) We define