Sequence of continuous functions vs. Lebesgue integration

[kennylcc001]

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!

 

[Pere Callahan]

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