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What would be an example of a sequence of functions which shows that the restriction to spaces of finite measure in Egoroff's theorem is necessary?

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    Just a suggestion: If you're going to keep posting so many questions in quick succession, all of which smell like homework then it would be a very nice treat if you shared a few thoughts on the matters. This might increase the willingness of people to spend some time answering your questions dramatically.2011-09-15

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Consider $f_n(x) = \chi_{[n, n+1]}(x)$, where $\chi_A$ is the indicator (characteristic) function of $A$.