- I was wondering what theorem(s) makes possible exchanging the order of Lebesgue integrals, for instance, in the following example: $$\int\nolimits_0^1 \int_0^x \quad 1 \quad dy dx = \int_0^1 \int_y^1 \quad 1 \quad dx dy,$$ or more generally $$\int_0^1 \int_0^x \quad f(x,y) \quad dy dx = \int_0^1 \int_y^1 \quad f(x,y) \quad dx dy.$$ I am not sure if it is Fubini's theorem because I have questions regarding it in the next part.
In Fubini's theorem:
- Must the set over which the double/overall integral is taken be a "rectangle" subset, i.e. $I_1 \times I_2$, instead of a general subset in the product space?
- Must the set over which the inner integral is taken not depend on the dummy variable in the outer integral?
Thanks and regards!