Very difficult Real Analysis question on Lebesgue integration

[bbkrsen585]

Suppose g_n are nonnegative and integrable on [0, 1], and that g_n $$\rightarrow$$ g almost everywhere.

Further suppose that for all $$\epsilon$$ > 0, $$\exists$$ $$\delta$$ > 0 such that for all A $$\subset$$ [0, 1], we have

meas(A) < $$\delta$$ implies that sup_n $$\int$$_A |g_n| < $$\epsilon$$.

Prove that g is integrable, and that $$\int$$_[0,1] g = lim $$\int$$_[0,1] g_n.

Attempt at the solution:
There are some observations I've compiled. First, for an set A with measure less than $$\delta$$, we know that the the limiting function g exists on that set. So, it's integral is also bounded by the sup. But from here, I don't know how to argue that the limiting function has a uniform bound on all of [0, 1].

 

[snipez90]

Have you tried any of the well-known theorems in Lebesgue integration regarding interchange of limit and integral?