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I've been trying to find an example for when a function converges almost surely but not mean square convergence and vice versa but I cant seem to find any easy examples on the web so I'm basically asking if any of you have some simple examples to help my understanding with them.

K B
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  • Relevant: http://math.stackexchange.com/questions/167218/why-does-l2-convergence-not-imply-almost-sure-convergence (for convergence in $L^2$ yet not a.s.) – Clement C. Mar 14 '16 at 18:21
  • Also http://math.stackexchange.com/questions/138043/does-convergence-in-lp-implies-convergence-almost-everywhere –  Mar 14 '16 at 18:22

1 Answers1

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The sequence $f_n(x)=n1_{(0,\frac{1}{n})}$ converges to zero almost everywhere, but not in $L^2([0,1])$.

carmichael561
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