Need some kind of convergence theorem for integrals taken over sequences of sets

[benorin]

I think this be Analysis,
I Need some kind of convergence theorem for integrals taken over sequences of sets, know one? Example, a double integral taken over sets such that
x^(2n)+y^(2n)<=1 with some integrand. I'd be interested in when the limit of the integral over the sequence of sets is equal to the integral over the limit of the sequence of sets (the unit square in the example).

 

[Samy_A]

I think this be Analysis,
I Need some kind of convergence theorem for integrals taken over sequences of sets, know one? Example, a double integral taken over sets such that
x^(2n)+y^(2n)<=1 with some integrand. I'd be interested in when the limit of the integral over the sequence of sets is equal to the integral over the limit of the sequence of sets (the unit square in the example).

Dominated convergence theorem or Vitali convergence theorem can be used.

If $(S_n)_n$ are the sets, and they "converge" to $S$, then you can set $\displaystyle f_n=\chi_{S_n}f$..
Still not sure whether we also need continuity or boundedness of f.

May depend on how you define the convergence of the sequence of sets.