Fubini's theorem

Fubini's theorem is an important theorem in the integral calculus. It specifies under what conditions, and how to calculate multidimensional integrals with the help of one-dimensional integrals. For the first time this set of Guido Fubini was proved.

Description

Using the Riemann integral or the Lebesgue integral, one can define the integration of functions over multidimensional areas. The problem here is that these integrals have been defined above a limit value by means of a cutting of the area into smaller parts. However, this does not result in useful, constructive method to calculate such integrals. In one-dimensional integrals can avoid this limit formation when the function to be integrated to find an antiderivative can ( Fundamental Theorem of Calculus ).

With the help of the theorem of Fubini multidimensional integrals can now be attributed to one-dimensional, which in turn with the help of a primitive function ( if known ) can be calculated. The theorem states also that the order of the one-dimensional integrations does not matter. This trick has been used in a naive way (before an exact definition of the integration calculation) in the 16th century and is in the special case of volume calculations known as the principle of Cavalieri.

Fubini's theorem for the Riemann integral

Be steady and I and J are compact intervals.

Then is continuous and it is

Fubini's theorem for the Lebesgue integral

And are two finite measure spaces and a measurable function, which is integrated with respect to the product dimension, i.e. it applies

Or it was necessary almost everywhere.

Then for almost all the function

And for almost all the function

Integrated or non-negative. It may therefore be the functions defined by integration or by

Consider. These are also integrable or non-negative and it is

Set of Tonelli (also Fubini - Tonelli )

A useful variant of this last sentence is the set of Tonelli. Here, the integration degree as a condition with respect to the product is not needed. It is enough that the iterated integrals exist for:

Be a real measurable function as above. If one of the two iterated integrals

Exists, then there is also the other, the product dimension is integrable with respect and we have:

Conclusions

By component-wise examination shows immediately that Fubini's theorem is true not only for real-valued functions, but also for according to functions with values ​​in finite-dimensional real vector spaces. Since the field of complex numbers is a two dimensional vector space, Fubini's theorem also applies to complex-valued functions, or functions with values ​​in finite - dimensional vector spaces

Using the theorem of Fubini can prove the following identities, which are used for example in the stochastics.

• Be Lebesgue integrable, then:

• Be Lebesgue integrable, then inductively follows: