- #1
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 [itex](X,\Sigma,\mu)[/itex] be a measure space.
Suppose that [itex] (h_n)_{n=1}^\infty[/itex] is a sequence of non-negative-real-valued integrable functions and that
[itex]
\lim_{n\to \infty}\int_Xh_nd\mu=0.
[/itex]
Is it true that the functions [itex]h_n[/itex] converge pointwise to [itex]0[/itex] almost-everywhere?
The Attempt at a Solution
My intuition is that since the [itex]h_n[/itex] 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!