Show L^p(E) is separable for any measurable E.

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Discussion Overview

The discussion revolves around the separability of the space L^p(E) for any measurable set E, focusing on the existence of a countable dense subset. Participants explore various approaches and considerations related to this topic, including the use of step functions and simple functions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that step functions over intervals with rational endpoints could serve as a countable dense subset in L^p(E), but express uncertainty regarding the inclusion of all intervals in E.
  • Another participant suggests finding an embedding from L^p(E) into L^p(ℝ) as a potential approach.
  • There is a clarification regarding the terminology, with one participant questioning whether "step functions" should be referred to as "simple functions."
  • Some participants note that the separability does not hold for p = infinity, indicating that L^∞ is not separable.
  • A later reply mentions the possibility of using simple functions with rational coefficients as part of an argument, suggesting that this might be found in graduate analysis textbooks.

Areas of Agreement / Disagreement

Participants generally agree that L^∞ is not separable, but there are multiple competing views regarding the construction of a dense subset for L^p(E) when p < ∞, and the discussion remains unresolved.

Contextual Notes

There are limitations regarding the assumptions about the measurable set E and the specific properties of the functions being discussed, which may affect the validity of the proposed approaches.

jpriori
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I have a sense that the countable, dense subset I'm looking for is the step functions, maybe over intervals with rational endpoints, but I'm not sure how to deal with the fact that E is any L-msb set, so there's no guarantee all the intervals are in there.
 
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You could find an embedding from ##L^p(E)## into ##L^p(\mathbb{R})##.
 
jpriori said:
I have a sense that the countable, dense subset I'm looking for is the step functions, maybe over intervals with rational endpoints, but I'm not sure how to deal with the fact that E is any L-msb set, so there's no guarantee all the intervals are in there.

Don't you mean the simple functions?
 
Though it isn't true if p = infinity, is it?
 
Robert1986 said:
Though it isn't true if p = infinity, is it?

You're right, L^oo is not separable.

Edit: I think there is an argument using simple functions with rational coefficients. It should
be in most graduate Analysis books.
 
Last edited:

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