Basic Inequalities of Lebesgue Integrals

Question

I have two very basic questions on Lebesgue integrals:

Q1) I know that if $f$ and $g$ are measurable and if $0\leq g\leq f$ on $E$, then $\int_E g\leq\int_E f$. My question is why is the condition $0\leq g$ needed? (or is it unnecessary? any counterexamples?)

Q2) What is the difference and relationship between $$\int_Ef_k\to\int_E f$$ and $$\int_E |f_k-f|\to 0$$? From what I understand, the second seems to be stronger, I can intuitively see that $f_k-f\leq|f_k-f|$ so if the second tends to zero, so should the first. Any counterexamples here that the first does not imply the second?

Thanks for any help.

Answer 1

1. The assumption $0\le g$ assures that both integrals exist (in $[0,\infty]$).

2. $f_k-f\le |f_k-f|$ doesn't help you. But $|\int f_k - \int f| = |\int (f_k -f)| \le \int|f_k-f|$ does help. Useful example: Let $f_k(x) = \sin (kx), f(x) = 0$ on $[0,1].$