- 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?
The answers to the above two questions seem to be "must" and "must not", based on Wikipedia and Planetmath.
Thanks and regards!