Continuity of the integral

  • #1

Main Question or Discussion Point

Hi,

I was wondering if the Lebesgue-integral is continuos with respect to the [itex]L^\infty[/itex] norm.

More precisely, assume there is a space of functions

[tex]
\mathcal{P}=\{f\in L^1 :||f||_{L^1}=1\}\cap L^\infty
[/tex]

endowed with the essential supremum norm [itex]||\cdot||_{L^\infty}[/itex]. If there is then a Cauchy sequence [itex]f_1,f_2,\dots[/itex] in [itex]\mathcal{P}[/itex], can one conclude that the limit (which exists in [itex]L^\infty[/itex]) is also integrable with [itex]L^1[/itex] norm one?

Thank you very much.
Pere
 

Answers and Replies

  • #2
morphism
Science Advisor
Homework Helper
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How about something like [itex]f_n = \frac{1}{n} \chi_{[0,n]}[/itex] in [itex]L^1(\mathbb{R})[/itex]? Here we have [itex]\|f_n \|_{L^1} = 1[/itex] and [itex]\|f_n - 0\|_{L^\infty} = \frac{1}{n} \to 0[/itex], while [itex]\| 0 \|_{L^1} = 0[/itex].
 
  • #3
Thanks morphism, that seems to answer my question.
 

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