I have the following $\{f_n\}^\infty$ sequence of smooth functions where $f_n:[0,1] \to \Re$ and $f_n(0) = 0$ with the following assumptions:
$$ f_n(x) \to f(x)\ \forall x \in [0,1] $$ $$ f_n' \to g \mbox{ a.e.} $$ $$ \int_{0}^{1} \lvert f_n'(x)\rvert^2dx \le 1$$
How do I prove $\{f_n'\}$ is uniformly integrable (find a $\delta$ such that for any set within $[0,1]$ with measure less than $\delta$ the integral over that set for each $f_n$ is less than $\epsilon$)?
I also need to prove $f(x) = \int_{0}^{x} g(t)dt$.
My solution:
From Hölder's inequality we know that $f_n \in L^1$ but can't understand how to use the rest of the information.
The uniform integrability of the family $\{f'_n\}$ follows from Cauchy-Schwarz inequality: $$\int_A\left|f'_n(x)\right|\mathrm dx\leqslant\left(\int_0^1\left|f'_n(x)\right|^2\mathrm dx\right)^{1/2}\lambda(A)^{1/2}\leqslant \lambda(A)^{1/2}.$$ For the second part, start from $$f_n(x)=f_n(0)+\int_0^xf'_n(t)\mathrm dt$$ and use the fact that a uniformly integrable sequence which converges pointwise to a function also converges in $\mathbb L^1$.