Lebesgue integral and convergence?

Question

I got stuck on this problem, maybe someone can give me a hint

Let (X, M , m) be a measure space and let (f_n: X --> [0, oo]) be a
sequence of functions that converges to a function f. Suppose that lim (\Integral(over X) f_n dm) = \Integral(over X) f dm < oo. Then, for every set E in the sigma-algebra M, we have lim (\Integral(over E) f_n dm) = \Integral(over E) f dm. Show that this conclusion may fail if \Integral(over X) f dm = oo.

If lim (\Integral(over E) f_n dm) = \Integral(over E) f dm is not true, then it's not hard to show that there's a subsequence (f_n_k) such that lim (\Integral(over E) f_n_k dm) < \Integral(over E) f dm, but I don't see why this leads to a contadiction.

Thank you

Answer 1

http://mathworld.wolfram.com/LebesgueIntegral.html
good luck

Answer 2

Hint: Fatou.