projections of analytic sets are analytic

Projections along compact paved spaces

Given sets X and K, the projection map πX:X×KX is defined by πX(x,y)=x. An important property of analytic setsMathworldPlanetmath ( is that they are stable under projections.

Theorem 1.

Let (X,F) be a paved space (, (K,K) be a compactPlanetmathPlanetmath ( paved space and πX:X×KX be the projection map.

If SX×K is F×K-analytic then πX(S) is F-analytic.

The proof of this follows easily from the definition of analytic sets. First, there is a compact paved space (K,𝒦) and a set T(×𝒦×𝒦)σδ such that S=πX×K(T). Then,


However, (K×K,𝒦×𝒦) is a compact paved space (see products of compact pavings are compact (, which shows that πX(S) satisfies the definition of -analytic sets.

Projections along Polish spaces

Theorem 1 above can be used to prove the following result for projections from the productMathworldPlanetmathPlanetmathPlanetmath of a measurable spaceMathworldPlanetmathPlanetmath and a Polish spaceMathworldPlanetmath. For σ-algebrasMathworldPlanetmath ( and , we use the notation for the product σ-algebra (, in order to distinguish it from the product paving ×.

Theorem 2.

Let (X,F) be a measurable space and Y be a Polish space with Borel σ-algebra ( B.

If SX×Y is FB-analytic, then its projection onto X is F-analytic.

An immediate consequence of this is the measurable projection theorem.

Although Theorem 2 applies to arbitrary Polish spaces, it is enough to just consider the case where Y is the space of real numbers with the standard topology. Indeed, all Polish spaces are Borel isomorphic to either the real numbers or a discrete subset of the reals (see Polish spaces up to Borel isomorphism), so the general case follows from this.

If Y=, then the Borel σ-algebra is generated by the compact paving 𝒦 of closed and bounded intervals. The collectionMathworldPlanetmath a(×𝒦) of analytic sets is closed underPlanetmathPlanetmath countableMathworldPlanetmath unions and countable intersectionsMathworldPlanetmath so, by the monotone class theorem, includes the product σ-algebra . Then, as the analytic sets define a closure operatorPlanetmathPlanetmathPlanetmath,


Thus every -analytic set is ×𝒦-analytic, and the result follows from Theorem 1.

Title projections of analytic sets are analytic
Canonical name ProjectionsOfAnalyticSetsAreAnalytic
Date of creation 2013-03-22 18:46:21
Last modified on 2013-03-22 18:46:21
Owner gel (22282)
Last modified by gel (22282)
Numerical id 8
Author gel (22282)
Entry type Theorem
Classification msc 28A05
Related topic MeasurableProjectionTheorem