I am using Munkres' Analysis on Manifolds textbook. Munkres defines Fubini's Theorem on rectangles and on simple regions (at least till the point that I have read).
Now, according to the book, we cannot use Fubini's Theorem all the time because it is quite possible that Integral over a region exists but the iterated integral does not (because of problems with either of the single integrals), or the iterated integral exists but the function cannot be integrated over the region.
From the exercises of the chapters, it seems that I need to apply Fubini's Theorem on rectifiable sets as opposed to simple regions and rectangles. My question is that if we know with certainty that the Integral exists over the region and we know that each of the single integrals in the iterated integral exist, then is it always true that the integral over the region will equal the iterated integral?
Best Answer
For $X,Y$ $\sigma$-finite measure spaces and a measurable function, $f:X\times Y\to\mathbb{R}$, as long as any one of the following three integrals are finite, then all three integrals, without the absolute value bars, exist and are equal.
$$\int_X\left(\int_Y |f(x,y)|\,\text{d}y\right)\,\text{d}x$$ $$\int_Y\left(\int_X |f(x,y)|\,\text{d}x\right)\,\text{d}y$$ $$\int_{X\times Y} |f(x,y)|\,\text{d}(x,y)$$
This might be known as the The Fubini–Tonelli theorem in your book, as the original theorem was a bit weaker.