What is the contradiction in the proof of Hahn Decomposition Theorem?

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The discussion centers on the contradiction found in the proof of the Hahn Decomposition Theorem, specifically regarding a positive subset G of E. The participants highlight that if G is a positive subset of E, then G being a subset of N implies that N cannot be a negative set, which contradicts the theorem's assumptions. The confusion arises from the interpretation of the relationships between the sets G, E, and N, particularly in demonstrating that G being a subset of P contradicts G being a subset of N.

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nateHI
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In the proof of the Hahn Decomposition Thm located here, there is the following sentence at the top of page 4.

"However, we shall prove that there is a positive subset of E carrying a positive charge, thereby obtaining the
contradiction."

but, if ##G## is the positive subset of ##E## mentioned above then
##
G \subset E\subset N
## which implies that ##N## is not a negative set. That seems to support the incorrect assumption, not contradict it. What am I missing?
 
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I haven't gone through the details, but I think it is trying to show that [itex]G\subset{P}[/itex] contradicting [itex]G\subset{N}[/itex].
 
mathman said:
I haven't gone through the details, but I think it is trying to show that [itex]G\subset{P}[/itex] contradicting [itex]G\subset{N}[/itex].
Yup. You're correct. Thanks, I was having a hard time with that one.
 

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