- Fubini's theorem
In
mathematical analysis , Fubini's theorem, named afterGuido Fubini , states that if:int_{A imes B} |f(x,y)|,d(x,y)
the integral being taken with respect to a
product measure on the space over A imes B, where "A" and "B" are complete measure spaces, then:int_Aleft(int_B f(x,y),dy ight),dx=int_Bleft(int_A f(x,y),dx ight),dy=int_{A imes B} f(x,y),d(x,y),
the first two integrals being iterated integrals with respect to two measures respectively, and the third being an integral with respect to a product of these two measures. Also,
:int_A f(x), dx int_B g(y), dy = int_{A imes B} f(x)g(y),d(x,y)
the third integral being with respect to a product measure.
If the above integral of the absolute value is not finite, then the two iterated integrals may actually have different values. See below for an illustration of this possibility.
Another version of Fubini's theorem states that if "A" and "B" are
sigma-finite measure spaces, not necessarily complete, and if either int_Aleft(int_B |f(x,y)|,dy ight),dxor int_Bleft(int_A |f(x,y)|,dx ight),dy , then int_{A imes B} |f(x,y)|,d(x,y) and int_Aleft(int_B f(x,y),dy ight),dx=int_Bleft(int_A f(x,y),dx ight),dy=int_{A imes B} f(x,y),d(x,y). In this version the condition that the measures are sigma-finite is necessary. Tonelli's theorem
Tonelli's theorem (named after
Leonida Tonelli ) is a successor of Fubini's theorem. The conclusion of Tonelli's theorem is identical to that of Fubini's theorem, but the assumptions are different. Tonelli's theorem states that on the product of two σ-finite measure spaces, a product measure integral can be evaluated by way of an iterated integral for "nonnegative" measurable functions, regardless of whether they have finite integral.In fact, the existence of the first integral above (the integral of the absolute value), can be guaranteed by Tonelli's theorem (see below).
A formal statement of Tonelli's theorem is identical to that of Fubini's theorem, except that the requirements are now that ("X", "A", μ) and ("Y", "B", ν) are σ-finite measure spaces, while "f" maps "X"×"Y" to [0, ∞] .
Applications
One of the most beautiful applications of Fubini's theorem is the evaluation of the
Gaussian integral which is the basis for much of probability theory::int_{-infty}^infty e^{-x^2},dx = sqrt{pi}.
To see how Fubini's theorem is used to prove this, see
Gaussian integral .Another nice use of Tonelli's theorem is to apply it to f(x,y)| for a complex valued function f .
It is useful to note that if
:varphi(x)=int |f(x,y)|,dy and int varphi(x), dx
then
:int |f(x,y)| , d(x,y) < infty.
This is often a useful way to check the conditions of Fubini's theorem.
Rearranging a conditionally convergent iterated integral
The iterated integral
:int_0^1int_0^1 frac{x^2-y^2}{(x^2+y^2)^2},dy,dx
does not converge absolutely (i.e. the integral of the
absolute value is not finite)::int_0^1int_0^1left|frac{x^2-y^2}{(x^2+y^2)^2} ight|,dy,dx=infty.
Fubini's theorem tells us that if the integral of the absolutevalue is finite, then the order of integration does not matter;if we integrate first with respect to "x" and then with respectto "y", we get the same result as if we integrate first withrespect to "y" and then with respect to "x". The assumptionthat the integral of the absolute value is finite is"Lebesgue integrability". That theassumption of Lebesgue integrability in Fubini's theoremcannot be dropped can be seen by examining this particulariterated integral. Clearly putting "dx" "dy" in placeof "dy" "dx" has the effect of multiplying the value ofthe integral by −1 because of the "antisymmetry" of thefunction being integrated. Therefore, unless the value of theintegral is zero, putting "dx" "dy" in place of"dy" "dx" actually changes the value of the integral.That is indeed what happens in this case.
Proof
One way to do this without using Fubini's theorem is as follows:
:int_0^1int_0^1left|frac{x^2-y^2}{(x^2+y^2)^2} ight|,dx,dy=int_0^1left [int_0^yfrac{y^2-x^2}{(x^2+y^2)^2},dx+int_y^1frac{x^2-y^2}{(x^2+y^2)^2},dx ight] ,dy:int_0^1left(frac{1}{2y}+frac{1}{2y}-frac{1}{y^2+1} ight),dy=int_0^1 frac{1}{y},dy-int_0^1frac{1}{1+y^2},dy.
Evaluation
Firstly, we consider the "inside" integral.
:int_0^1frac{x^2-y^2}{(x^2+y^2)^2},dy:int_0^1 frac{x^2 + y^2 - 2y^2}{(x^2 + y^2)^2} , dy:int_0^1 frac{1}{x^2 + y^2} , dy + int_0^1 frac{-2y^2}{(x^2 + y^2)^2} , dy:int_0^1 frac{1}{x^2 + y^2} , dy + int_0^1 y left(frac{d}{dy} frac{1}{x^2 + y^2} ight) , dy:int_0^1 frac{1}{x^2 + y^2} , dy + left(left [frac{y}{x^2 + y^2} ight] _{y=0}^1 - int_0^1 frac{1}{x^2 + y^2} , dy ight) (by parts):frac{1}{1 + x^2}.
This takes care of the "inside" integral with respect to "y";now we do the "outside" integral with respect to "x":
:int_0^1frac{1}{1+x^2},dx=left [arctan(x) ight] _0^1
=arctan(1)-arctan(0)=frac{pi}{4}.
Thus we have
:int_0^1int_0^1frac{x^2-y^2}{(x^2+y^2)^2},dy,dx=frac{pi}{4}
and
:int_0^1int_0^1frac{x^2-y^2}{(x^2+y^2)^2},dx,dy=-frac{pi}{4}.
Fubini's theorem implies that since these two iterated integrals differ, the integral of the absolute value must be ∞.
tatement
When
:int_a^bint_c^d left|f(x,y) ight|,dy,dx=infty
then the two iterated integrals
:int_a^bint_c^d f(x,y),dy,dx mbox{and} int_c^dint_a^b f(x,y),dx,dy
may have different finite values.
Strong versions of Fubini's theorem
The existence of strengthenings of
Fubini's theorem , where the function is no longer assumed to be measurable but merely that the two iterated integrals are well defined and exist, is independent of the standardZermelo–Fraenkel axioms ofset theory .Martin's axiom implies that there exists a function on the unit square whose iterated integrals are not equal, while a variant ofFreiling's axiom of symmetry implies that in fact a strong Fubini-type theorem for [0, 1] does hold, and whenever the two iterated integrals exist they are equal [Chris Freiling, "Axioms of symmetry: throwing darts at the real number line", J. Symbolic Logic 51 (1986), no. 1, 190–200.] . SeeList of statements undecidable in ZFC .References
Wikimedia Foundation. 2010.