How to prove that if function is integrable then the integral as a function of the set is continuous ?
We assume that the function is nonnegative and function going from arbitrary measure space with nonatomic measure. The topology: on sigma algebra induced by semi-metric where $d(A,B)$= symmetric difference operation.
Hint: It is enough to show that if $A_n$ is a sequence of sets such that the measure of $A_n$ tends to $0$, then $\int_{A_n} f(x)\,\mathrm{d}x\to 0$