Re: L^2 functions question...

hmm...
I think Fatou's lemma is enough...It tells us that f is in L^2 and
||f||_2 <= 1

The generalization of your problem is a weak convergence theorem.

Suppse 1<p<infty, f_k\in L^p(E) and f_k -> f, ||f_k||_p <= M. Then for
any g in L^q (where p, q are conjugate index), we have
lim \int_E f_k(x)g(x)dx = \int_E f(x)g(x)dx.

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