Convergence of Lebesgue measurable sets I've been working on the following result:
Let $f$ be Lebesgue measurable on $[0,1]$ with $f(x)>0$ almost everywhere on $[0,1]$. Assume there are measurable sets $E_k \in [0,1]$ with $\int_{E_k} f(x)\to 0$ as $k \to \infty$. Then $m(E_k) \to 0$ as $k \to \infty$.
I've been attempting to bound $f$ in some way as then the result follows quickly, but it doesn't seem that I can do this as the values of $f$ range over $[0,1]$.

How about: Suppose $m(E_k)$ doesn't tend to zero. (The following holds a.e.)
$$
\int_{E_k} f(x) \ dx = \int_{[0,1]} \chi(E_k) f(x) \ dx > \int_{[0,1]} \chi(E_k) \ dx = m(E_k)
$$
with the inequality since $f > 0$ a.e. on $[0,1]$. Then take limits as $k \to \infty$ to derive a contradiction.
