Can you give an example of a sequence of non-negative functions tending to zero pointwise such their integral tends to zero but there is no integrable function which bounds them?
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The following works $$f_n=n1_{[0,1/n^2]} $$
frusstu
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I think this is a valid example, but it would improve the Answer to explain why there is no (upper) bounding function of the sequence that is integrable. – hardmath Mar 30 '15 at 17:07