Analytical amount

Analytical sets are considered in the mathematical sub-areas of measure theory and descriptive set theory ; they are special subsets of Polish spaces . They are more general than Borel quantities , but still have certain measurability properties.

definition

A subset of a Polish space is called analytical if there is a Polish space and a continuous mapping with . In short: Analytical quantities are constant images of Polish spaces. ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle Z}$ ${\displaystyle f\colon Z\rightarrow X}$ ${\displaystyle f(Z)=A}$

The empty set should also be analytical. Therefore, one must either allow the empty set as Polish space or explicitly accept the empty set.

properties

• Countable unions and countable averages of analytical sets are again analytical.

• Complements of analytical sets are generally not again analytical.

• In a Polish area every Borel set is analytical; the converse is generally not true.

• Analytical sets have the Baire property .

• Every analytical set is Lebesgue measurable .

Projections of Borel quantities

Analytical quantities can be characterized as projections of Borel quantities as follows. For two sets and let be the projection onto the second component. The following statements are then equivalent for a subset of a Polish area: ${\displaystyle X}$ ${\displaystyle Z}$ ${\displaystyle \pi _{2}\colon Z\times X\rightarrow X}$ ${\displaystyle A\subset X}$

1. ${\displaystyle A}$ is analytical.
2. There is a Polish room and a locked crowd with . ${\displaystyle Z}$ ${\displaystyle C\subset Z\times X}$ ${\displaystyle A=\pi _{2}(C)}$
3. There's a Polish room and a Borel crowd with me . ${\displaystyle Z}$ ${\displaystyle B\subset Z\times X}$ ${\displaystyle A=\pi _{2}(B)}$

To prove this, it suffices to consider the case that is not empty. Is analytical, by definition, is for a continuous function on Polish space . Then the graph is finished and shows the end from 1. to 2.. Since closed sets are Borel sets, 3. follows from 2. Finally, if 3. there is a Polish space and a continuous mapping with , because Borel sets are analytical. Then there is a continuous picture of a Polish area and therefore analytical. ${\displaystyle A\subset X}$ ${\displaystyle A}$ ${\displaystyle A=f(Z)}$ ${\displaystyle f\colon Z\rightarrow X}$ ${\displaystyle Z}$ ${\displaystyle G(f)\subset Z\times X}$ ${\displaystyle \pi _{2}(G(f))=f(Z)=A}$ ${\displaystyle Y}$ ${\displaystyle f\colon Y\rightarrow Z\times X}$ ${\displaystyle B=f(Y)}$ ${\displaystyle A=(\pi _{2}\circ f)(Y)}$

Separation theorem for analytical quantities

The following separation theorem for analytical sets goes back to NN Lusin :

• Let there be a Polish space and two disjoint analytic sets. Then there are two disjoint Borel sets with and . ${\displaystyle X}$ ${\displaystyle A_{1},\,A_{2}\subset X}$ ${\displaystyle B_{1},\,B_{2}\subset X}$ ${\displaystyle A_{1}\subset B_{1}}$ ${\displaystyle A_{2}\subset B_{2}}$

Conclusion: An analytic set is a Borel set if and only if the complement is also analytic. ${\displaystyle A\subset X}$ ${\displaystyle X\setminus A}$

To prove the consequence, let us first consider a Borel set. Then it is also Borel set and therefore analytical. Conversely, if analytically, so turn above separation theorem on and on. Because of the disjointness it must then be, that is, is a Borel set. ${\displaystyle A}$ ${\displaystyle X\setminus A}$ ${\displaystyle X\setminus A}$ ${\displaystyle A_{1}=A}$ ${\displaystyle A_{2}=X\setminus A}$ ${\displaystyle A_{1}=B_{1}}$ ${\displaystyle A}$

The Baire room

A special Polish room is the Baire room with the product topology . is the space of all sequences of natural numbers, the topology is generated, for example, from the complete metric defined by , where is the smallest index on which the two sequences differ. One can show that every (non-empty) Polish space is a continuous picture of . From the definition of the analytical set it follows immediately: ${\displaystyle {\mathcal {N}}:=\mathbb {N} ^{\infty }}$ ${\displaystyle {\mathcal {N}}}$ ${\displaystyle {\vec {n}}=(n_{k})_{k}}$ ${\displaystyle d({\vec {n}},{\vec {m}}):=2^{-n({\vec {n}},{\vec {m}})}}$ ${\displaystyle n({\vec {n}},{\vec {m}})}$ ${\displaystyle {\mathcal {N}}}$

• A non-empty subset of a Polish space is analytical if and only if there is a continuous mapping with . ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle f\colon {\mathcal {N}}\rightarrow X}$ ${\displaystyle f({\mathcal {N}})=A}$

By means of space one can obtain all analytic sets of a Polish space as a projection of a fixed analytic set. The following sentence applies: ${\displaystyle {\mathcal {N}}}$

• Be a Polish room. Then there is an analytic subset such that ${\displaystyle X}$ ${\displaystyle A\subset {\mathcal {N}}\times X}$

${\displaystyle \{x\in X;\,({\vec {n}},x)\in A\},\quad {\vec {n}}\in {\mathcal {N}}}$

exactly the analytical quantities of passes through. ${\displaystyle X}$

If one applies this theorem to, one can show that there is an analytic set in that is not a Borel set. ${\displaystyle X={\mathcal {N}}}$ ${\displaystyle \{{\vec {n}};\,({\vec {n}},{\vec {n}})\in A\}}$ ${\displaystyle {\mathcal {N}}}$

In the case of the Baire space, every analytical set can already be represented as a projection of a closed set in ; in the case of the real numbers and the Cantor space , projections of countable sections of open sets in or are sufficient . ${\displaystyle {\mathcal {N}}^{2}}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle {\mathcal {C}}^{2}}$

Universal measurability

A subset of a measurement space is universally measurable , if for every finite measure on amounts available with and . Any amount out is universally measurable, because in this case one can choose. Obviously the set of all universally measurable sets forms a σ-algebra which, according to what has just been said, includes the σ-algebra . ${\displaystyle T}$ ${\displaystyle (X,{\mathcal {X}})}$ ${\displaystyle \mu }$ ${\displaystyle (X,{\mathcal {X}})}$ ${\displaystyle B_{1},\,B_{2}\in {\mathcal {X}}}$ ${\displaystyle B_{1}\subset T\subset B_{2}}$ ${\displaystyle \mu (B_{2}\setminus B_{1})=0}$ ${\displaystyle {\mathcal {X}}}$ ${\displaystyle B_{1}=T=B_{2}}$ ${\displaystyle {\mathcal {X}}}$

Polish spaces are naturally measurement spaces by providing them with the σ-algebra of Borel sets, and with regard to this measurement space, universal measurability in Polish spaces is to be understood. Then:

• Every analytical quantity of a Polish area is universally measurable.

In particular, every analytical set is Lebesgue measurable . Since there are analytical sets that are not Borel sets, the σ-algebra of the universally measurable sets is in general genuinely larger than the σ-algebra of the Borel sets.

Cuts

If a surjective mapping is called a mapping a section of , if . The existence of such a mapping follows easily from the axiom of choice , in that one chooses and sets a prototype for each by means of surjectivity . If and measuring rooms and is measurable, then the question arises whether one can find a measurable cut . ${\displaystyle f\colon X\rightarrow Y}$ ${\displaystyle g\colon Y\rightarrow X}$ ${\displaystyle f}$ ${\displaystyle f\circ g=\mathrm {id} _{Y}}$ ${\displaystyle y\in Y}$ ${\displaystyle x_{y}\in f^{-1}(\{y\})}$ ${\displaystyle g(y)=x_{y}}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle f}$ ${\displaystyle g}$

To investigate this question, we call a measurement space countably separated , if there is a sequence of sets from , so that for every two different points from one can always be found that contains exactly one of the two points. This is called an analytical Borel space if it as a measuring space isomorphic to a measuring space , wherein an analytic subset of a Polish space and the σ-algebra of the averages of the amounts of Borel with is. With these terms, the following sentence applies: ${\displaystyle (Y,{\mathcal {Y}})}$ ${\displaystyle (E_{n})_{n}}$ ${\displaystyle {\mathcal {Y}}}$ ${\displaystyle Y}$ ${\displaystyle E_{n}}$ ${\displaystyle (Y,{\mathcal {Y}})}$ ${\displaystyle (A,{\mathcal {A}})}$ ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle X}$ ${\displaystyle A}$

• Let there be an analytical Borel room, a countably separated measuring room and a measurable image. Then there is a - -measurable cut of , where the σ-algebra of the universally measurable sets is. ${\displaystyle (X,{\mathcal {X}})}$ ${\displaystyle (Y,{\mathcal {Y}})}$ ${\displaystyle f\colon X\rightarrow Y}$ ${\displaystyle {\mathcal {U}}}$ ${\displaystyle {\mathcal {X}}}$ ${\displaystyle f}$ ${\displaystyle {\mathcal {U}}}$ ${\displaystyle (Y,{\mathcal {Y}})}$

Such sentences play a decisive role in the structure and representation theory of type IC * algebras , as explained in the textbook by W. Arveson given below , or in the disintegration of Von Neumann algebras , such as can be found in .

Historical remark

In a publication from 1905, H. Lebesgue was wrongly of the opinion that he had shown that the projection of a Borel set of the plane onto the axis was again a Borel set. In 1917 MJ Suslin had discovered the error contained therein, introduced the analytic sets and showed that there are analytic sets that are not Borel sets. ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle x}$

See also

• Projective hierarchy , the analytic (and co-analytic) sets form the first level of the projective hierarchy.

Individual evidence

1. ^ Donald L. Cohn: Measure Theory. Birkhäuser, Boston MA et al. 1980, ISBN 3-7643-3003-1 , chapter 8.2.
2. ^ Kazimierz Kuratowski : Topology. Volume 1. New edition, revised and augmented. Academic Press, New York et al. 1966, ISBN 0-1242-9201-1 , p. 485.
3. ^ Donald L. Cohn: Measure Theory. Birkhäuser, Boston MA et al. 1980, ISBN 3-7643-3003-1 , Theorem 8.3.1.
4. ^ Donald L. Cohn: Measure Theory. Birkhäuser, Boston MA et al. 1980, ISBN 3-7643-3003-1 , sentence 8.2.16.
5. ^ Donald A. Martin , Descriptive Set Theory: Projective Sets. In: Jon Barwise (Ed.): Handbook of Mathematical Logic (= Studies in Logic and the Foundations of Mathematics. Vol. 90). North-Holland, Amsterdam et al. 1977, ISBN 0-7204-2285-X , pp. 783-815, here p. 790, doi : 10.1016 / S0049-237X (08) 71121-2 .
6. ^ Donald L. Cohn: Measure Theory. Birkhäuser, Boston MA et al. 1980, ISBN 3-7643-3003-1 , Corollary 8.4.3
7. ^ William Arveson : Invitation to C * -algebras (= Graduate Texts in Mathematics. Vol. 39). Springer, New York NY et al. 1976, ISBN 0-387-90176-0 , Theorem 3.4.3.
8. ^ William Arveson: Invitation to C * -algebras (= Graduate Texts in Mathematics. Vol. 39). Springer, New York NY et al. 1976, ISBN 0-387-90176-0 , chapter 4
9. ↑ Gert K. Pedersen: C * -Algebras and their Automorphism Groups (= LMS Monographs. Vol. 14). Academic Press Inc., London et al. 1979, ISBN 0-12-549450-5 , chapter 4.
10. ^ Richard M. Dudley: Real Analysis and Probability (= Cambridge Studies in Advanced Mathematics. Vol. 74). Cambridge University Press, Cambridge et al. 2002, ISBN 0-521-00754-2 , p. 500.