# Who first proved Fubini's theorem $n$th order integrals?

Who first proved a generalized Fubini theorem for integrals of order $$≥3$$?

An $$n$$th order integral is $$\underbrace{\underset{x_n}\int\underset{x_{n-1}}\int\ldots\underset{x_1}\int}_{n} f(x_1,x_2,\ldots,x_n)dx_1dx_2\ldots dx_n.$$

By "a generalized Fubini theorem," I mean that all permutations of the $$x_i$$ produce valid and equivalent integrals.

• Can you clarify your sense of "order $\ge$ 3"? Commented Oct 8, 2019 at 21:10
• @paulgarrett See what I added. Commented Oct 8, 2019 at 21:35
• Odd -- I would have thought that any proof for order 2 would suffice to prove for order N. Just break into a double-integral inside or outside of the remaining integrals. We may need some commentary from math.SE here Commented Oct 9, 2019 at 14:31
• It is a trivial consequence of order 2 and induction. You can get any permutation by transposing pairs. Commented Oct 9, 2019 at 22:00
• @Conifold There was no commentary on math.SE on this question (as you can see). Anway, an inductive argument should be sufficient. Commented Oct 10, 2019 at 13:17