An Application of Fubini-Tonelli Theorem

Question

In a question here, selected answer is started with an application of Fubini's theorem. which I can't figure it out at all. I'm having difficulties with figuring out the differential elements. such as the equation $$ \int_{0}^{a}\int_0^{f(t)}\mathrm{d}s\mathrm{d}t = \int_{0}^{f(a)}\int_{f^{-1}(s)}^a \mathrm{d}t\mathrm{d}s $$ and a similar one that's been written for $ \int_{0}^{f(a)}\int_0^{f^{-1}(t)}\mathrm{d}s\mathrm{d}t $. Plus I don't understand, how the upper and lower limits have been changed when differential elements $ \text ds $ and $ \text dt $ are switched.

Answer 1

The answer you are referring to is integrating the function identically equal to $1$, it is usually implicitly the case when we are integrating "nothing", maybe it is clearer this way : let $g(s)=1$ $\forall s $ $$ \int_{0}^{a} \int_{0}^{f(t)} g(s)\text ds\text dt +\int_{0}^{f(a)} \int_{0}^{f^{-1}(t)} g(s)\text ds\text dt \\$$ $$= \int_{0}^{a} \int_{0}^{f(t)} 1\text ds\text dt +\int_{0}^{f(a)} \int_{0}^{f^{-1}(t)} 1\text ds\text dt$$

$$=\int_{0}^{a} f(t) dt +\int_{0}^{f(a)} f^{-1}(t)dt$$

where the last equality comes from the fact that $$ \int_{0}^{f(t)} 1\text ds=f(t)$$