Resnick's exercise 3.15 asks the following:
Measurable indicator function that cannot be approximated by a continuous sequence $f_n$
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measure-theory
probability-theory
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0For the secondary question, see http://math.stackexchange.com/questions/15088/is-every-lebesgue-measurable-function-on-mathbbr-the-pointwise-limit-of-con – 2012-06-06
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1Regarding the formatting: use the `cases` environment. – 2012-06-06