Sequence of continuous functions vs. Lebesgue integration

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SUMMARY

The discussion revolves around constructing a sequence of continuous functions \( f_n : [0,1] \to [0,\infty) \) that converge pointwise to zero while satisfying specific integral conditions. The proposed solution involves using peaked functions that become narrower and higher as \( n \) increases, resembling a delta sequence. The key conditions are that \( \int_0^1 f_n \, dx \to 0 \) and \( \int_0^1 \sup f_n \, dx = \infty \). This construction effectively demonstrates the nuances of Lebesgue integration in relation to pointwise convergence.

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This is a question from Papa Rudin Chapter 2:

Find continuous functions f_{n} : [0,1] -> [0,\infty) such that f_{n} (x) -> 0 for all x \in [0 ,1] as $n -> \infty. \int^{1}_{0} f_n dx -> 0 , but \int^{1}_{0} sup f_{n} dx = \infty.

Any idea? :) Thank you so much!
 
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Try some sort of peaked functions, whose peaks get higher and narrower (and move in some way in [0,1]) as n increases. I'm thinking of something like a delta sequence. these should converge pointwise to zero. Humm, the integral should converge to zero...for ordinary delta sequences the integral is always one, so you might want to make even narrower. the third condidtion should also be satisfied. Maybe I'll try to write it down more explicitly later:smile:
 
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