Since uniformly we have that for all there exists an such that implies now taking we get for all and this last one is integrable since so picking we get our dominating function.
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Just stuck on a Measure theory question right now:
Suppose uniformly. Show that if and . Find an example that shows that the result is not necessarily true when
Little bit stumped on how to start. Initially I thought I could use the dominated convergence theorem, but not fully sure. We do have uniform convergence and the dominated convergence theorem claims if we can bound a function by g then the .
If not I'm thinking of using chebychev's inequality.
Any help would be great.
I also apologise for any problems with the LaTex, it's my first time using it.
Thanks