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