gauss mouse
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Homework Statement
This question is not the assignment problem but I think that if the result I mention here is true, then my assignment problem will be solved.
Let (X,\Sigma,\mu) be a measure space.
Suppose that (h_n)_{n=1}^\infty is a sequence of non-negative-real-valued integrable functions and that
<br /> \lim_{n\to \infty}\int_Xh_nd\mu=0.<br />
Is it true that the functions h_n converge pointwise to 0 almost-everywhere?
The Attempt at a Solution
My intuition is that since the h_n are non-negative functions, we can rule out the possibility of cancellation but I don't know what else to do and maybe the result is not even true anyway.
I'd really appreciate any help!