Proof of Boundedness for Sets with Measure Zero?

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Homework Help Overview

The discussion revolves around the concept of boundedness in relation to Jordan measurable sets and their boundaries having measure zero. Participants are exploring the implications of these properties in mathematical analysis.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are examining the relationship between a set's boundary having measure zero and the boundedness of the set itself. There is a request for a proof regarding this relationship, along with a clarification about the conditions under which Jordan measure applies.

Discussion Status

The discussion is active, with participants questioning the assumptions related to boundedness and measure zero. Some guidance has been provided regarding the applicability of Jordan measure to bounded sets, but there is no explicit consensus on the broader implications for sets with measure zero boundaries.

Contextual Notes

There is a mention of specific definitions regarding measure zero and the conditions for Jordan measurability, which may influence the interpretation of the problem. The original poster's request for a proof indicates a need for deeper exploration of these concepts.

ak416
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I saw this come up in a proof: Since A is a Jordan measurable set (bd(A) has measure zero), there exists a closed rectangle B s.t A subset of B. So basically theyre saying, if bd(A) has measure zero then A is bounded. Can someone give me a quick proof of that? By the way when i say a set S has measure zero i mean for every e>0 there is a cover {U1,U2,...} of S by rectangles s.t. sum(i=1 to infinity)(volume(Ui)) < e.
 
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Jordan measure only applies to bounded sets.
 
o ok my bad. But in general, if you have any set with its boundary being of measure zero, it doesn't necessarily mean its bounded right?
 
You are correct.
 

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