Analytic set

Analytical amounts are considered in the mathematical sciences of measure theory and descriptive set theory, there are special subsets of Polish spaces. They are more general than Borel sets, but still have some Messbarkeitseigenschaften.

Definition

A subset of a Polish space is called analytic if there is a Polish space and a continuous map with. In short: Analytical quantities are continuous images of Polish spaces.

The empty set is to be analytical. So you either need to allow the empty set as a Polish space or explicitly accept the empty set.

Properties

• Countable unions and countable intersections of analytic sets are analytic back.

• Complements of analytical amounts are generally not re- analysis.

• In a Polish space every Borel set is analytic, the converse is not true in general.

• Analytic sets have the Baire property.

• Every analytic set is Lebesgue measurable.

Projections of Borel sets

Analytical quantities can be characterized as projections of Borel sets as follows. Two quantities and is the projection on the second component. For a subset of a Polish space then the following statements are equivalent:

For the proof it suffices to consider the case that is not empty. Is analytic, then by definition, for a continuous function on a Polish space. Then the graph is complete and, so the conclusion would shown by 1 by 2. Since closed sets are Borel sets, it follows from 3 2 Is Finally 3 before, then there's a Polish space and a continuous map, for Borel sets are analytic. Then the continuous image of a Polish space and hence is analytic.

Separation theorem for analytic sets

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

• There are a Polish space and two disjoint analytic sets. Then there are two disjoint Borel sets and.

Conclusion: An analytical quantity is exactly then a Borel set if the complement is analytic.

To prove the first implication is Borel set. Then is also Borel set and therefore analytically. Conversely, analytically, so turn the above separation theorem on and on. Because of the disjunction must then be, that is a Borel set.

The Baire space

A special Polish space is Baire space with the product topology. is the space of all consequences of natural numbers, the topology is generated, for example by the full -defined metric with the smallest index, at which the two sequences differ. One can show that every ( non-empty ) Polish space is a continuous image of. From the definition of the analytic set therefore results directly:

• A non - empty subset of a Polish space is then exactly analytic if there is a continuous map with.

By means of the room you can get all analytical quantities of a Polish space as a projection of a solid analytic set. We have the following sentence:

• Be a Polish space. Then there is an analytic subset such that

Exactly passes through the analytical quantities of.

Applying this rate to where it can be shown that an analytic set in, which is not a Borel set.

In the case of Baire space, each analytic set rich projections countable sections of open sets can already be as a projection of a closed set in the present, in the case of real numbers and the Cantor space in the or.

Universal measurability

A subset of a measurement space is called universally measurable if, for every finite measure on quantities and. Each set of universally measurable, because in this case one can choose. Apparently, the set of all universally measurable amounts of a σ - algebra, which includes just what is said after the σ - algebra.

Polish spaces are naturally measure spaces by providing them with the σ - algebra of Borel sets, and with respect to this Maßraums is to understand universal measurability in Polish spaces. Then:

• Every analytic set a Polish space is universally measurable.

In particular, so every analytic set is Lebesgue measurable. Since there are analytical amounts no Borel quantities σ algebra of universal measurable amounts is generally strictly greater than the σ - algebra of Borel quantities.

Slice

Is a surjective map, it is called a mapping a cut of if. The existence of such a map follows easily from the axiom of choice by adding to each one chooses an archetype means surjectivity and sets. Are and measure spaces and measurable, so the question arises whether one can find a measurable section.

To study this question we call a measure space countably separated if there is a sequence of sets in, so that for any two different points always can be found that contains exactly one of the two points. This is called an analytic Borel space if it is isomorphic as a measure space to a measure space, where an analytic subset of a Polish space and the σ -algebra of Borel sets of the averages with. With these terms, the following sentence is true:

• Let an analytic Borel space, a countable Hausdorff measure space and a measurable map. Then there is a - measurable section of, the σ - algebra of universally measurable quantities would respect.

Such sentences play a crucial role in the structure and representation theory of type IC *-algebras, as in the below textbook by W. Arveson is executed, or in the disintegration of von Neumann algebras, as they can be found for example in.

Historical Note

H. Lebesgue was in a publication from 1905, erroneously, to have shown that the projection of a Borel set is a Borel set of the plane back on the x -axis. MJ Suslin in 1917 uncovered the error contained therein, introduced the analytical quantities and shown that there are analytic sets which are not Borel sets.