Vector measure

A vectorial measure is a term from the measure theory. It represents a generalization of the Maßbegriffes: The measure is no longer real-valued, but vector-valued. Vector dimensions are used in applications of functional analysis used ( spectral measure ).

Definition

A vectorial dimension is a finite or countably -additive -valued set function, that is:

Let be a measurable space (ie a non-empty set and a σ - algebra) and a Banach space with norm. A -valued set function to a function with.

The function is called finitely additive, if finally applies many, pairwise disjoint sets of for.

The function is called countably additive (also - additive) if

For every sequence of pairwise disjoint sets, where the convergence of the sum on the right side in the Banach space to be understood. Since the quantities for each sequence of pairwise disjoint from to apply and since any rearrangement of such a sequence whose union and thus the left side of the above formula does not change, the sum on the right side with rearrangements must remain unchanged; that is, it is automatically unconditional convergence before.

Total variation

Analogous to the signed dimensions, one can also introduce the total variation of a vector measure: It is a -valued set function. The total variation of the function

By

Is explained. These are from a set and a measurable partition of a partition of which consists of quantities from. One can show that the total variation of a finitely or countably additive, positive measure is, if is finite or countably additive. A vectorial measure is of bounded variation if its total variation is finite, that is, when. Some authors, such as Serge Lang, understand by vectorial measurements only those of bounded variation. Here we follow the terminology of Diestel - Uhl, in the vectorial dimensions need not be of bounded variation. We have the following sentence:

• Banach space is finite-dimensional, then the total variation of a finite extent, which means that it is of limited variation.

In infinite-dimensional Banach spaces is a vectorial measure not necessary of bounded variation. As an example, the half-line is the Borel sets, is the consequence space. For his, the Lebesgue measure is on. Then a vectorial measure with values ​​in, that is not of bounded variation.

The space of countably additive measures of bounded variation with values ​​in the Banach space is clearly a vector space. With the total variation as norm becomes a Banach space.

Examples

• Any complex or signed measure is a vectorial measure.
• Each spectral measure defines a finitely additive measure vectorial.
• Let the unit interval and the - algebra of Lebesgue measurable amounts of. For in denote the characteristic function of. Depending on the choice of the value range of different vector dimensions are hereby defined: The function is a finitely additive measure vectorial, which is not countably additive and not of bounded variation.
• The function is a countably additive measure vectorial.

The Radon - Nikodym

Be a measurable space, a positive measure on a Banach space and. Then by

A vectorial measure

Defined.

A vectorial measure is called -continuous or absolutely against continuous if and always follows. Easy, one can show that the above defined absolutely continuous is against. Be

Then is a closed subspace of. The Radon - Nikodym theorem is concerned with the question whether every -continuous vector measure is already on the form. As a generalization of the classical theorem of Radon - Nikodym one obtains:

Be a σ - finite, positive measure on the measuring space is a Hilbert space. Then the mapping, an isometric isomorphism. In particular, each continuous - level vector based on the shape, wherein, uniquely determined.