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

In summary, the conversation discusses the proof of the Hahn Decomposition Theorem and the concept of a positive subset of a set. The contradiction arises from the assumption that a set is both negative and positive, leading to the conclusion that it is not negative. The conversation ultimately concludes that the proof is trying to show that the positive subset is a subset of a positive set, contradicting the assumption that it is a subset of a negative set.
  • #1
nateHI
146
4
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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  • #2
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].
 
  • #3
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.
 

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

What is the Hahn Decomposition Proof?

The Hahn Decomposition Proof is a mathematical theorem that states that any measurable function can be decomposed into two parts: a positive part and a negative part. This provides a way to separate positive and negative values within a function.

Who discovered the Hahn Decomposition Proof?

The Hahn Decomposition Proof was discovered by the mathematician Hans Hahn in the early 20th century.

What are the applications of the Hahn Decomposition Proof?

The Hahn Decomposition Proof has various applications in mathematics, particularly in measure theory and functional analysis. It is also used in economics, game theory, and probability theory.

What are the key concepts in the Hahn Decomposition Proof?

The key concepts in the Hahn Decomposition Proof are the Radon-Nikodym theorem, the Lebesgue integral, and the notion of measurable sets and functions.

Is the Hahn Decomposition Proof unique?

No, there are multiple versions of the Hahn Decomposition Proof that have been developed over the years. However, all versions share the same fundamental concept of separating a measurable function into positive and negative parts.

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