L^1 convergence and uniform convergence

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Discussion Overview

The discussion revolves around the concepts of L^1 convergence and uniform convergence within the context of measure theory. Participants explore examples of sequences of functions that converge uniformly but do not converge in the L^1 norm, particularly focusing on the implications of using different measure spaces.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant seeks an example of a measure space (X,µ) and a sequence {f_n} of L^1 functions that converge uniformly to a function f, but not in L^1(X,µ).
  • Another participant argues that if the measure space is Borel, a sequence of Borel measurable functions that converges must converge to a Borel function, suggesting the original statement may not hold true.
  • A later reply provides an example using X=R with Lebesgue measure, describing a sequence of rectangles that converge uniformly to 0 but not in L^1, and suggests that it can be arranged so that the sequence does not converge in any L^p space.
  • There is a clarification regarding the definition of L^1 as the space of integrable functions, questioning why the limit function f=0 is not considered in this space.
  • Participants confirm that f(x) = 0 is indeed in L^1, but the focus remains on whether the sequence f_n converges to f in the L^1 norm.
  • One participant asks for clarification on the definition of the L^1 norm.

Areas of Agreement / Disagreement

Participants express differing views on the validity of the original statement regarding convergence in Borel spaces. While some provide examples supporting the original claim, others challenge its validity, leading to an unresolved debate on the topic.

Contextual Notes

The discussion includes assumptions about the properties of Borel measurable functions and the nature of convergence in different L^p spaces, which remain unresolved.

quasar987
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Let (X,µ) be a measure space with µ(X) infinite. I'm trying to find an example of a pair (X,µ) and a sequence {f_n} of L^1(X,µ) functions converging uniformly to a function f such that we do not have f_n --> f in L^1(X,µ).
 
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If your measure space is Borel then I don't think this statement is true. A sequence of Borel measurable functions that has a limit converges to a Borel function.
 
RedX said:
If your measure space is Borel then I don't think this statement is true. A sequence of Borel measurable functions that has a limit converges to a Borel function.
I don't see how you can conclude the statement is false from this.

In fact it is possible to come up with an example satisfying the OP using X=R with Lebesgue measure. Take f_n to be a sequence of rectangles with decreasing height but increasing width, in such a way that f_n -> 0 uniformly but not in L^1. You can even rig it so that f_n doesn't converge in any L^p (1 <= p < inf).
 
morphism said:
In fact it is possible to come up with an example satisfying the OP using X=R with Lebesgue measure. Take f_n to be a sequence of rectangles with decreasing height but increasing width, in such a way that f_n -> 0 uniformly but not in L^1. You can even rig it so that f_n doesn't converge in any L^p (1 <= p < inf).

Doesn't L^1 mean the space of integrable functions? So why isn't f=0 in this space?
 
RedX said:
Doesn't L^1 mean the space of integrable functions? So why isn't f=0 in this space?

Yes, f(x) = 0 is in L^1. The issue is whether f_n converges to f in the L^1 norm.
 
Ah! Thank you morphism.
 
kilimanjaro said:
Yes, f(x) = 0 is in L^1. The issue is whether f_n converges to f in the L^1 norm.

What is the L^1 norm?
 
The L^1 norm is defined on the space of all integrable functions as
\| f\|_{L^1(X,\mu)}=\int_X |f|d\mu<br />
 

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