measure and limit

[eugene]

Let (A_n)_n be a sequence of Lebesgue measurable subsets of [0,1], such that 
for any interval (a,b)  we have that 
lim_{n->infty} m (A_n & (a;b) ) = (b-a)/3.
Prove that for any function f: [0,1]->R of bounded variation lim_{n->infty} 
int_{A_n} f(x)dx = int_0^1 f(x)dx.

Here m is a Lebesgue measure on R and a sign "&" means an interesection.

Any ideas would be very apprecitable

Thanks