In multivariable calculus, this weaker result is sometimes also called Fubini's theorem, although it was already known by Leonhard Euler.
[3] A special case of Fubini's theorem for continuous functions on the product of closed, bounded subsets of real vector spaces was known to Leonhard Euler in the 18th century.
In 1904, Henri Lebesgue extended this result to bounded measurable functions on a product of intervals.
[4] Levi conjectured that the theorem could be extended to functions that are integrable rather than bounded[citation needed] and this was proven by Fubini in 1907.
The maximal product measure can be constructed by applying Carathéodory's extension theorem to the additive function μ such that μ(A × B) = μ1(A)μ2(B) on the ring of sets generated by products of measurable sets.
Fubini's theorem has some rather technical extensions to the case when X and Y are not assumed to be σ-finite (Fremlin 2003).
The crux of the theorem is that the interchange of order of summation holds even if the series diverges.
Without the condition that the measure spaces are σ-finite, all three of these integrals can have different values.
Another way is to add the condition that the support of f is contained in a countable union of products of sets of finite measures.
Fremlin (2003) gives some rather technical extensions of Tonelli's theorem to some non σ-finite spaces.
; these forms include Tonelli's theorem as a special case as the negative part of a non-negative function is zero and so has finite integral.
As in Fubini's theorem, the single integrals may fail to be defined on a measure 0 set.
The versions of Fubini's and Tonelli's theorems above do not apply to integration on the product of the real line
Most proofs involve building up to the full theorems by proving them for increasingly complicated functions, with the steps as follows.
This shows that Tonelli's theorem can fail for spaces that are not σ-finite no matter which product measure is chosen.
Fubini's theorem holds for spaces even if they are not assumed to be σ-finite provided one uses the maximal product measure.
If f is the characteristic function of E then the two iterated integrals of f are defined and have different values 1 and 0.
Assuming the continuum hypothesis, one can identify X with the unit interval I, so there is a bounded non-negative function on I×I whose two iterated integrals (using Lebesgue measure) are both defined but unequal.
[7] The stronger versions of Fubini's theorem on a product of two unit intervals with Lebesgue measure, where the function is no longer assumed to be measurable but merely that the two iterated integrals are well defined and exist, are independent of the standard Zermelo–Fraenkel axioms of set theory.
The continuum hypothesis and Martin's axiom both imply that there exists a function on the unit square whose iterated integrals are not equal, while Harvey Friedman (1980) showed that it is consistent with ZFC that a strong Fubini-type theorem for [0,1] does hold, and whenever the two iterated integrals exist they are equal.
When applied correctly, Fubini's theorem leads directly to an antiderivative function that can be integrated in an elementary way, which is shown in cyan in the following equation chain:
The value η(2) is equal to π²/12 and this can be proven with Fubini's theorem[dubious – discuss] in this way:
The original antiderivative, shown here in cyan, leads directly to the value of η(2): The improper integral of the Complete Elliptic Integral of first kind K takes the value of twice the Catalan constant accurately.
The Catalan constant can only be obtained via the Arctangent Integral, which results from the application of Fubini's theorem:
This time, the expression now in royal cyan color tone is not elementary, but it leads directly to the equally non-elementary value of the "Catalan constant" using the Arctangent Integral, also called Inverse Tangent Integral.
The same procedure also works for the Complete Elliptic Integral of the second kind E in the following way:
In this way it is shown accurately by using the Fubini's Theorem twice that these integrals are indeed identical to each other.
Now this formula for the squaring of an integral is set up: This chain of equations can then be generated accordingly:
For the Dilogarithm of one this value appears: In this way the Basel problem can be solved.
For the Lemniscatic special case of Legendre's relation, this result emerges: