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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

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# Continuity of the integral

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