Hi All, I've been having great difficulty making progress on this problem.
Suppose gn converges to g a.e. on [0,1]. And, for all n, gn and h are integrable over [0,1]. And |gn|[tex]\leq[/tex]h for all n.
Define Gn(x)=[tex]\int[/tex]gn(x) from 0 to x.
Define G(x)=[tex]\int[/tex]g(x) from 0 to x.
Prove: Gn converges to G uniformly.
2. The attempt at a solution
So, here's what I've got. We can use dominated convergence theorems to show that Gn goes to G pointwise on [0,1]. Since the set is compact, this should also provide some insights.
Moreover, I was thinking we could use Egorov's theorem to take away the set on which G does not go to G uniformly. I'm not sure how to get rid of that otherwise.