Vectorial measure

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

Definitions

Vector measures are finite or countable additive set functions with values ​​in a Banach space. A measurement space (i.e. a non-empty set and a σ-algebra ) and a Banach space are more precise . A -valent set function on is a function . One calls a finitely additive measure if ${\displaystyle (\Omega ,\Sigma )}$ ${\displaystyle E}$ ${\displaystyle E}$ ${\displaystyle \Sigma }$ ${\displaystyle \nu \colon \Sigma \rightarrow E}$ ${\displaystyle \nu }$

${\displaystyle \nu (\varnothing )=0}$ and

${\displaystyle \nu (A_{1}\cup \cdots \cup A_{n})=\nu (A_{1})+\cdots +\nu (A_{n})}$

for a finite number of pairwise disjoint sets from true. One speaks of a countable-additive measure, if ${\displaystyle A_{1},\ldots ,A_{n}}$ ${\displaystyle \Sigma }$

${\displaystyle \nu (\varnothing )=0}$ and

${\displaystyle \nu {\bigg (}\biguplus _{n=1}^{\infty }A_{n}{\bigg )}=\sum _{n=1}^{\infty }\nu (A_{n})}$

for every sequence of pairwise disjoint sets , where the convergence of the sum on the right is to be understood in Banach space . Since this should apply to every sequence of pairwise disjoint sets from and since any rearrangement of such a sequence does not change its union and thus the left side of the above formula, the sum on the right side must also remain unchanged during rearrangements; that is, there is automatically unconditional convergence . ${\displaystyle (A_{n})_{n\in \mathbb {N} }}$ ${\displaystyle A_{n}\in \Sigma }$ ${\displaystyle E}$ ${\displaystyle \Sigma }$

When we talk about a measure, we mean a countable-additive measure. Let it be the set of all -value measures on the measuring room . If there are two such measures and is a scalar, then are through ${\displaystyle M(\Omega ,\Sigma ,E)}$ ${\displaystyle E}$ ${\displaystyle (\Omega ,\Sigma )}$ ${\displaystyle \nu _{1},\nu _{2}}$ ${\displaystyle \alpha }$

${\displaystyle (\nu _{1}+\nu _{2})(A):=\nu _{1}(A)+\nu _{2}(A)}$

${\displaystyle (\alpha \cdot \nu _{1})(A):=\alpha \cdot \nu _{1}(A)}$

Dimensions and from where. The operations so defined make a vector space. ${\displaystyle \nu _{1}+\nu _{2}}$ ${\displaystyle \alpha \cdot \nu _{1}}$ ${\displaystyle M(\Omega ,\Sigma ,E)}$ ${\displaystyle M(\Omega ,\Sigma ,E)}$

Is , then we get the space of scalar measures , which becomes a Banach space with the total variation norm . The attempt to transfer this to spaces of vectorial dimensions comes up against an obstacle. The generalized total variation is not automatically finite for all measures, but this can be cured by the concept of semi-variation. ${\displaystyle E=\mathbb {R} }$ ${\displaystyle M(\Omega ,\Sigma )=M(\Omega ,\Sigma ,\mathbb {R} )}$ ${\displaystyle \|\nu \|:=|\nu |(\Omega )}$

Total variation

Analogous to the signed measures , one can also introduce the total variation of a vector measure: Let it be a -value set function. The total variation of is the function ${\displaystyle \nu }$ ${\displaystyle E}$ ${\displaystyle \nu }$

${\displaystyle \|\nu \|\colon \Sigma \rightarrow [0,\infty ],}$

by

${\displaystyle \|\nu \|(A):=\sup {\bigg \{}\sum _{n=1}^{\infty }\|\nu (A_{n})\|\,|\,(A_{n})_{n\in \mathbb {N} }{\text{ ist messbare Zerlegung von }}A{\bigg \}}}$

is explained. Here a set from and a measurable decomposition of a partition of , which consists of sets from . One can show that the total variation of a finite or countable additive, positive measure is when finite or countable additive. A vectorial measure is of limited variation if its total variation is finite, that is, if . Some authors, e.g. B. Serge Lang, understand by vectorial measures only those of limited variation. Here we follow the terminology of Diestel-Uhl, in which vector dimensions need not be of limited variation. The following sentence applies: ${\displaystyle A}$ ${\displaystyle \Sigma }$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle \Sigma }$ ${\displaystyle \nu }$ ${\displaystyle \nu }$ ${\displaystyle \|\nu \|(\Omega )<\infty }$

• If the Banach space is finite-dimensional, then the total variation of is a finite measure, that is, it is of limited variation. ${\displaystyle E}$ ${\displaystyle \nu }$ ${\displaystyle \nu }$

In infinite-dimensional Banach spaces, a vectorial measure is not necessarily of limited variation. As an example, the half-line with the Borel sets , is the sequence space . For be , where the Lebesgue measure is on . Then there is a vectorial measure with values ​​in that is not of limited variation. ${\displaystyle (\Omega ,\Sigma )}$ ${\displaystyle \mathbb {R} _{0}^{+}=[0,\infty )}$ ${\displaystyle E}$ ${\displaystyle \ell ^{2}}$ ${\displaystyle A\in \Sigma }$ ${\displaystyle \nu (A):=({\frac {1}{n}}\lambda (A\cap [n-1,n]))_{n\in \mathbb {N} }\in \ell ^{2}}$ ${\displaystyle \lambda }$ ${\displaystyle [0,\infty )}$ ${\displaystyle \nu }$ ${\displaystyle \ell ^{2}}$

The space of the countable additive measures of limited variation with values ​​in the Banach space is a subspace of . With total variation as the norm, it becomes a Banach space. ${\displaystyle M^{1}(\Omega ,\Sigma ,E)}$ ${\displaystyle E}$ ${\displaystyle M(\Omega ,\Sigma ,E)}$ ${\displaystyle \|\nu \|:=\|\nu \|(\Omega )}$ ${\displaystyle M^{1}(\Omega ,\Sigma ,E)}$

Semivariation

The semivariation of a vectorial measure presented here eliminates the disadvantage of the total variation of not always being finite. This is bought at the price of not always getting a countable additive measure. Let there be a measuring space, a Banach space and a vectorial measure. It is easy to consider that for each of the dual space there is a scalar measure . In this way we get a linear operator ${\displaystyle (\Omega ,\Sigma )}$ ${\displaystyle E}$ ${\displaystyle \nu \colon \Sigma \rightarrow E}$ ${\displaystyle \varphi \circ \nu }$ ${\displaystyle \varphi }$ ${\displaystyle E'}$ ${\displaystyle (\Omega ,\Sigma )}$

${\displaystyle T_{\nu }\colon E'\rightarrow M(\Omega ,\Sigma ),\quad T_{\nu }(\varphi ):=\varphi \circ \nu }$

into the Banach space of scalar measures . With the help of the theorem of the closed graph one shows that is even bounded . In order to define an image by ${\displaystyle M(\Omega ,\Sigma )}$ ${\displaystyle (\Omega ,\Sigma )}$ ${\displaystyle T_{\nu }}$ ${\displaystyle \Sigma \rightarrow \mathbb {R} }$

${\displaystyle |\nu |_{\infty }(A):=\sup\{|\varphi \circ \nu |(A);\,\varphi \in E',\|\varphi \|\leq 1\}}$

and names the semivariation of . Because of ${\displaystyle |\nu |_{\infty }(\cdot )\colon \Sigma \rightarrow \mathbb {R} }$ ${\displaystyle \nu }$

${\displaystyle |\nu |_{\infty }(A)\leq |\nu |_{\infty }(\Omega )=\sup\{\|T_{\nu }(\varphi )\|;\,\varphi \in E',\|\varphi \|\leq 1\}=\|T_{\nu }\|}$

this quantity is always finite, but the semivariation is generally only a monotonic, countable subadditive set function. becomes with the norm ${\displaystyle M(\Omega ,\Sigma ,E)}$

${\displaystyle \|\nu \|_{\infty }:=|\nu |_{\infty }(\Omega )=\|T_{\nu }\|}$

a Banach room.

Examples

• Each complex or signed dimension is a vector dimension.
• Each spectral measure defines a finite additive vectorial measure.
• Let it be the unit interval and the algebra of the Lebesgue measurable sets of . For in denote the characteristic function of . Depending on the choice of the value range, different vector dimensions are defined: ${\displaystyle \Omega }$ ${\displaystyle [0,1]}$ ${\displaystyle \Sigma }$ ${\displaystyle \sigma }$ ${\displaystyle \Omega }$ ${\displaystyle A}$ ${\displaystyle \Sigma }$ ${\displaystyle \nu (A):=\chi _{A}}$ ${\displaystyle A}$
  • The function is a finite additive vectorial measure that is not countable additive and not of limited variation. ${\displaystyle \nu \colon \Sigma \rightarrow L^{\infty }[0,1]}$
  • The function is a countable additive vectorial measure. ${\displaystyle \nu \colon \Sigma \rightarrow L^{1}[0,1]}$
• Let it be and be the sequence space of the null sequences. Choose a solid and defining the vectorial measure by ${\displaystyle (\Omega ,\Sigma )=(\mathbb {N} ,{\mathcal {P}}(\mathbb {N} ))}$ ${\displaystyle E}$ ${\displaystyle c_{0}}$ ${\displaystyle (\alpha _{n})_{n}\in c_{0}}$ ${\displaystyle \nu \colon {\mathcal {P}}(\mathbb {N} )\rightarrow c_{0}}$

${\displaystyle \nu (A):=\sum _{n\in A}\alpha _{n}e_{n}}$,

where is the sequence that has a 1 in the nth place and only zeros otherwise. The following applies to the total variation ${\displaystyle e_{n}\in c_{0}}$

${\displaystyle \|\nu \|(A)=\sum _{n\in A}|\alpha _{n}|}$

and for the semivariation one gets

${\displaystyle |\nu |_{\infty }(A)=\sup _{n\in A}|\alpha _{n}|}$.

The integral according to a vectorial measure

As above, let it be a measuring space, a Banach space and a vectorial measure. Next, let the Banach space of limited, measurable functions (or ) with the supremum norm . We want the integral ${\displaystyle (\Omega ,\Sigma )}$ ${\displaystyle E}$ ${\displaystyle \nu \colon \Sigma \rightarrow E}$ ${\displaystyle B(\Omega ,\Sigma )}$ ${\displaystyle \Omega \rightarrow \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \|\cdot \|_{\infty }}$

${\displaystyle \int _{\Omega }f\mathrm {d} \nu \in E}$

explain for functions . Each defines a continuous, linear functional ${\displaystyle f\in B(\Omega ,\Sigma )}$ ${\displaystyle f\in B(\Omega ,\Sigma )}$

${\displaystyle \psi _{f}\colon M(\Omega ,\Sigma )\rightarrow \mathbb {R} ,\quad \psi _{f}(\mu ):=\int _{\Omega }f\mathrm {d} \mu }$.

Note that in this definition only the integral with respect to a scalar measure occurs, which is assumed to be known at this point.

We recall the operator introduced above

${\displaystyle T_{\nu }\colon E'\rightarrow M(\Omega ,\Sigma ),\quad T_{\nu }(\varphi ):=\varphi \circ \nu }$.

and consider the adjoint operator

${\displaystyle T_{\nu }'\colon M(\Omega ,\Sigma )'\rightarrow E''}$.

So we can apply this to and define it like this ${\displaystyle \psi _{f}}$

${\displaystyle \int _{\Omega }f\mathrm {d} \nu :=T_{\nu }'(\psi _{f})\in E''}$.

Finally, one considers that the integral defined in this way even lies in, whereby, as usual, by means of the canonical embedding in the binary space , one understands as a sub-vector space of . Since the simple functions are close to in , it suffices for a characteristic function to show that the above integral is actually in . Since the calculation required for this clarifies the above definitions, it should be carried out as an example of such an integral. Since above definition is an element off , we can apply it to any and get it ${\displaystyle E}$ ${\displaystyle E}$ ${\displaystyle E''}$ ${\displaystyle B(\Omega ,\Sigma )}$ ${\displaystyle f=\chi _{A}}$ ${\displaystyle E}$ ${\displaystyle E''}$ ${\displaystyle \varphi \in E'}$

${\displaystyle \int _{\Omega }\chi _{A}\mathrm {d} \nu (\varphi ):=T_{\nu }'(\psi _{\chi _{A}})(\varphi )=(\psi _{\chi _{A}}\circ T_{\nu })(\varphi )=\psi _{\chi _{A}}(T_{\nu }(\varphi ))}$

${\displaystyle =\psi _{\chi _{A}}(\varphi \circ \nu )=\int _{\Omega }\chi _{A}\mathrm {d} (\varphi \circ \nu )=(\varphi \circ \nu )(A)=\varphi (\nu (A))}$.

Since it was arbitrary, it follows ${\displaystyle \varphi \in E'}$

${\displaystyle \int _{\Omega }\chi _{A}\mathrm {d} \nu =\nu (A)}$

and that is actually an item . So the integral defined above is an element off for all bounded, measurable functions . This calculation also shows that one can use the expected formula for simple functions ${\displaystyle E}$ ${\displaystyle E}$

${\displaystyle \int _{\Omega }\sum _{k=1}^{n}\alpha _{k}\chi _{A_{k}}\,\mathrm {d} \nu \,=\,\sum _{k=1}^{n}\alpha _{k}\nu (A_{k})}$

receives. The integral over a measurable subset is then carried out as usual ${\displaystyle A\subset \Omega }$

${\displaystyle \int _{A}f\mathrm {d} \nu :=\int _{\Omega }\chi _{A}\cdot f\,\mathrm {d} \nu }$

Are defined. The following estimate applies:

${\displaystyle {\big \|}\int _{A}f\mathrm {d} \nu {\big \|}\leq \|f\|_{\infty }|\nu |_{\infty }(A)}$

for everyone . ${\displaystyle A\in \Sigma ,f\in B(\Omega ,\Sigma ),\nu \in M(\Omega ,\Sigma ,E)}$

Generalized theorems of mass theory

Yegorov's theorem

Egorov's classical theorem translates into vector dimensions as follows:

• Let there be a measuring space, a Banach space and a vectorial measure. Let it also be a sequence of measurable functions that converges point by point to a function . Then there is a measurable quantity for each with , so that the sequence converges to equally . ${\displaystyle (\Omega ,\Sigma )}$ ${\displaystyle E}$ ${\displaystyle \nu \colon \Sigma \rightarrow E}$ ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \Sigma }$ ${\displaystyle f}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle A\in \Sigma }$ ${\displaystyle |\nu |_{\infty }(A)<\varepsilon }$ ${\displaystyle \Omega \setminus A}$ ${\displaystyle f}$

Theorem of majorized convergence

The classical theorem of majorized convergence also applies to vector measures in the following form:

• Let there be a measuring space, a Banach space and a vectorial measure. Let it also be a uniformly bounded sequence of measurable functions that converges point by point to a function . Then converges ${\displaystyle (\Omega ,\Sigma )}$ ${\displaystyle E}$ ${\displaystyle \nu \colon \Sigma \rightarrow E}$ ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \Sigma }$ ${\displaystyle f}$

${\displaystyle \lim _{n\to \infty }\int _{\Omega }f_{n}\mathrm {d} \nu \,=\,\int _{\Omega }f\mathrm {d} \nu }$.

Radon-Nikodým theorem

The classical Radon-Nikodým theorem does not apply in full general to vectorial measures. For this purpose a measuring room, a positive dimension , a Banach room and . Then it's through ${\displaystyle (\Omega ,\Sigma )}$ ${\displaystyle \mu }$ ${\displaystyle \Sigma }$ ${\displaystyle E}$ ${\displaystyle f\in {\mathcal {L}}^{1}(\mu ,E)}$

${\displaystyle \mu _{f}(A)\,:=\,\int _{A}f\mathrm {d} \mu }$

a vectorial measure

${\displaystyle \mu _{f}\in M^{1}(\Omega ,\Sigma ,E)}$ With ${\displaystyle \|\mu _{f}\|=\|f\|_{1}}$

Are defined. Note that we are integrating a Banach space-valued function according to a scalar measure using the Bochner integral . In contrast to this, the integral for vector measures introduced above is explained for scalar-valued functions.

A vectorial measure is called -continuous or absolutely continuous against , if it follows from and always . It is easy to show that what is defined above is absolutely continuously against . Be ${\displaystyle \nu }$ ${\displaystyle \mu }$ ${\displaystyle \mu }$ ${\displaystyle A\in \Sigma }$ ${\displaystyle \mu (A)=0}$ ${\displaystyle \nu (A)=0\in E}$ ${\displaystyle \mu _{f}}$ ${\displaystyle \mu }$

${\displaystyle M^{1}(\mu ,E):=\{\nu \in M^{1}(\Omega ,\Sigma ,E);\,\nu {\mbox{ absolut stetig gegen }}\mu \}}$.

Then there is a closed subspace of . Radon-Nikodým's theorem deals with the question of whether every -continuous vectorial measure is already of form . Generalizing Radon-Nikodým's classical theorem, we get: ${\displaystyle M^{1}(\mu ,E)}$ ${\displaystyle M^{1}(\Omega ,\Sigma ,E)}$ ${\displaystyle \mu }$ ${\displaystyle \mu _{f}}$

• Let be a σ-finite , positive measure on the measurement space , let it be a Hilbert space . Then the mapping , an isometric isomorphism. In particular, every -continuous vectorial measure is of the form , with -being uniquely determined. ${\displaystyle \mu }$ ${\displaystyle (\Omega ,\Sigma )}$ ${\displaystyle E}$ ${\displaystyle L^{1}(\mu ,E)\rightarrow M^{1}(\mu ,E)}$ ${\displaystyle f\mapsto \mu _{f}}$ ${\displaystyle \mu }$ ${\displaystyle M^{1}(\Omega ,\Sigma ,E)}$ ${\displaystyle \mu _{f}}$ ${\displaystyle f\in {\mathcal {L}}^{1}(\mu ,E)}$ ${\displaystyle \mu }$

In addition to Hilbert spaces, there are also other Banach spaces that have an analogous property, these are called spaces with Radon-Nikodym property .

Tensor products

One possibility to construct functions with values ​​in a Banach space from scalar-valued functions is to use tensor products . It therefore makes sense to consider tensor products . Each ${\displaystyle M(\Omega ,\Sigma )\otimes E}$

${\displaystyle \nu =\sum _{k=1}^{n}\mu _{k}\otimes x_{k}\in M(\Omega ,\Sigma )\otimes E}$

is with the definition

${\displaystyle \nu (A):=\sum _{k=1}^{n}\mu _{k}(A)x_{k}}$

a vectorial measure. The various possibilities of normalizing such tensor products lead to the total variations or semivariations introduced above.

Projective tensor product

The norm of the projective tensor product coincides with the total variation, i.e. for each there is the total variation of the measure. In particular, all of these measures are of limited total variation and the completion is isometrically isomorphic to a subspace of . In general, this is a true sub-vector space, more precisely it is the sub-vector space of the vectorial measures with Radon-Nikodym property . ${\displaystyle \|\cdot \|_{\pi }}$ ${\displaystyle \nu =\sum _{k=1}^{n}\mu _{k}\otimes x_{k}\in M(\Omega ,\Sigma )\otimes E}$ ${\displaystyle \|\nu \|_{\pi }=\|\nu \|}$ ${\displaystyle M(\Omega ,\Sigma ){\hat {\otimes }}_{\pi }E}$ ${\displaystyle (M^{1}(\Omega ,\Sigma ,E),\|\cdot \|)}$

Injective tensor product

The norm of the injective tensor product coincides with the semivariation, that is, for each is the semivariation of the measure. The completion is isometrically isomorphic to a subspace of . In general, this is a true sub-vector space, more precisely it is the sub-vector space of the vectorial measures, the image set of which is relatively compact . ${\displaystyle \|\cdot \|_{\varepsilon }}$ ${\displaystyle \nu =\sum _{k=1}^{n}\mu _{k}\otimes x_{k}\in M(\Omega ,\Sigma )\otimes E}$ ${\displaystyle \|\nu \|_{\varepsilon }=\|\nu \|_{\infty }}$ ${\displaystyle M(\Omega ,\Sigma ){\hat {\otimes }}_{\varepsilon }E}$ ${\displaystyle (M(\Omega ,\Sigma ,E),\|\cdot \|_{\infty })}$ ${\displaystyle \nu (\Sigma )\subset E}$

Individual evidence

1. ^ Serge Lang: Real Analysis (= Addison-Wesley Series in Mathematics ). Addison-Wesley, Reading MA et al. 1969, ISBN 0-201-04179-0 .
2. ↑ J. Diestel, JJ Uhl Jr .: Vector measures. 1977.
3. ^ Serge Lang: Real Analysis (= Addison-Wesley Series in Mathematics ). Addison-Wesley, Reading MA et al. 1969, ISBN 0-201-04179-0 , XI, 4.5. Theorem 8.
4. ^ Serge Lang: Real Analysis (= Addison-Wesley Series in Mathematics ). Addison-Wesley, Reading MA et al. 1969, ISBN 0-201-04179-0 , XI, 4.5.
5. ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , sentence 5.3
6. ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , page 100 and sentence 5.10
7. ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , sentence 5.11
8. ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , sentence 5.12
9. ^ Serge Lang: Real Analysis (= Addison-Wesley Series in Mathematics ). Addison-Wesley, Reading MA et al. 1969, ISBN 0-201-04179-0 , XI, 4.5, Theorem 9.
10. ^ Serge Lang: Real Analysis (= Addison-Wesley Series in Mathematics ). Addison-Wesley, Reading MA et al. 1969, ISBN 0-201-04179-0 , XI, 4.5, Corollary 2 to Theorem 10.
11. ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , Theorem 5.22
12. ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , sentence 5.18

literature

• Joseph Diestel, John J. Uhl Jr .: Vector measures (= Mathematical Surveys. Vol. 15). American Mathematical Society, Providence RI 1977, ISBN 0-821-81515-6 .
• Serge Lang : Real and Functional Analysis (= Graduate Texts in Mathematics. Vol. 142). 3rd edition. Springer, New York NY et al. 1993, ISBN 0-387-94001-4 .
• Tsoy-Wo Ma: Banach-Hilbert Spaces, Vector Measures and Group Representations. World Scientific Publishing Company, River Edge NJ et al. 2002, ISBN 981-238-038-8 .
• Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1