Re: measure and limit

You left out a 3 in the last equation.

I'll rewrite it:

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 = (1/3)*int_0^1
f(x)dx.

Also I suspect you stated the hypotheses wrong - I
don't
see what this has to do with bounded variation.

I checked the statement, it is ok.
If you have a couterexample, you are welcome.
Thanks
.

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