In Real Analysis by Royden, chapter 11, section 4 proposition 17 why the domain of integration is not mentioned. In the above said text book chapter 11, section 4, proposition 17 why the domain of integration is not mentioned. 

I've got the second and third editions of Royden in front of me and in both editions Proposition 17 is a convergence theorem with the following conclusion:
$$\int f\,d\mu \le\liminf\int f_n\,d\mu_n$$
If this is the integral in question, then the domain of integration is the entire space, which is the convention when there is no other indication of a set over which the integration must take place.
