Prove that if $(f_n)$ is a sequence of nonnegative, measurable functions on $[a,b]$ such that $lim_{n\to\infty}\int_a^b f_n(x)dx=0$, then $(f_n)$ converges to $0$ in measure. Show by example that we cannot replace the conclusion with the assertion that $(f_n)$ converges to $0$ almost everywhere.
I don't really know how to go about proving this. I know that convergence almost everywhere implies convergence in measure and that if a sequence converges in measure then there exists a subsequence that converges almost everywhere but I haven't done a lot with convergence in measure.
Use Markov's inequality namely, $$ \mu(|f_n|>\varepsilon)\leq \frac{1}{\varepsilon}\int_a^b f_n(x)\, dx. $$