TITLE: Measure theory questions
QUESTION [8 upvotes]: i. If $1 < p < \infty$ and $E = \{f_a, a \in A\}$ set of measurable functions of $\mathbb{R}$ and $\sup_{a \in A} ||f_a||_p < \infty$, I want to show that for $ 0 < q < p$, $\lim \limits_{x \rightarrow \infty} \sup \limits_{a \in A} \int_{|f_a| > x} |f_a|^q = 0$.
ii. For bounded Borel $f$ on $\mathbb{R}$ where $\int fg = 0$ for all $g \in C_c(\mathbb{R})$, I want to show $ f=0$ a.e.
iii. I want to show that $\forall \epsilon\gt0, \exists \delta\gt0$ s.t. $m(E \subseteq \mathbb{R}) < \delta \Rightarrow \int_E |f_a|^q < \epsilon$ 
My thoughts were for one to use Hölder's inequality to  use Borel-Cantelli, but that didn't get me anywhere. For two, it seems obviously if $g \in C_c(\mathbb{R})$ then set of discontinuities are countable so then $f$ must be zero a.e.

REPLY [3 votes]: The answer to the first question has been already given. For the second one, 
first approximate pointwise the characteristic function of a compact set by continuous functions with compact support. 
As we can choose the sequence monotonic, we can actually show that for each compact $K$,
$$\int f\chi_Kd\lambda=0.$$
By regularity of Lebesgue measure, we can deduce that $\int f\chi_Sd\lambda=0$ for each measurable set $S$ of finite measure. Conclude.