analytic set

(Note: this entry concerns analytic setsMathworldPlanetmath as used in measure theory. For the definition in analytic spaces see analytic set (

For a continuous mapMathworldPlanetmath of topological spacesMathworldPlanetmath it is known that the preimagesMathworldPlanetmath of open sets are open, preimages of closed ( sets are closed and preimages of Borel sets are themselves Borel measurable. The situation is more difficult for direct imagesPlanetmathPlanetmath ( That is if f:XY is continuousMathworldPlanetmath then it does not follow that SX being open/closed/measurable implies the same property for f(S). In fact, f(X)=Image(f) need not even be measurable. One of the few things that can be said, however, is that f(S) is compactPlanetmathPlanetmath whenever S is compact. Analytic sets are defined in order to be stable under direct images, and their theory relies on the stability of compact sets. This is a fruitful concept because, as it turns out, all measurable setsMathworldPlanetmath are analyticPlanetmathPlanetmath and all analytic sets are universally measurable.

A subset S of a Polish spaceMathworldPlanetmath X is said to be analytic (or, a Suslin set) if it is the image of a continuous map f:ZX from another Polish space Z — see Cohn. It is then clear that g(S) will again be analytic for any continuous map g:XY between Polish spaces. Indeed, g(S) will be the image of gf.

Here, we instead give the following definition which applies to arbitrary paved spaces and, in particular, to all measurable spacesMathworldPlanetmath. Furthermore, for Polish spaces, it can be shown to be equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to the definition just mentioned above.

Recall that for a paved space (X,), σδ represents the collectionMathworldPlanetmath of countableMathworldPlanetmath intersectionsMathworldPlanetmath of countable unions of elements of , and that a paving is compact if every subcollection satisfying the finite intersection property has nonempty intersection.


Let (X,F) be a paved space. Then a set AX is said to be F-analytic, or analytic with respect to F, if there exists a compact paved space (K,K) and an S(F×K)σδ such that


Here, F×K is the product paving and π:X×KX is the projection map π(x,y)=x.

Writing this out explicitly, there are doubly indexed sequences of sets Am,n and Km,n𝒦 such that



A={xX:(x,y)S for some yK}.

Although this allows for 𝒦 to be any compact paving it can be shown that it makes no differencePlanetmathPlanetmath if it is just taken to be the collection of compact subsets of, for example, the real numbers.

For a measurable space (X,), a subset of X is simply said to be analytic if it is -analytic, and a subset of a topological space is said to be analytic if it is analytic with respect to the Borel σ-algebra (


  • 1 K. Bichteler, Stochastic integration with jumps. Encyclopedia of Mathematics and its Applications, 89. Cambridge University Press, 2002.
  • 2 Donald L. Cohn, Measure theory. Birkhäuser, 1980.
  • 3 Claude Dellacherie, Paul-André Meyer, Probabilities and potential. North-Holland Mathematics Studies, 29. North-Holland Publishing Co., 1978.
  • 4 Sheng-we He, Jia-gang Wang, Jia-an Yan,Semimartingale theory and stochastic calculus. Kexue Chubanshe (Science Press), CRC Press, 1992.
  • 5 M.M. Rao, Measure theory and integration. Second edition. Monographs and Textbooks in Pure and Applied Mathematics, 265. Marcel Dekker Inc., 2004.
Title analytic set
Canonical name AnalyticSet1
Date of creation 2013-03-22 18:44:52
Last modified on 2013-03-22 18:44:52
Owner gel (22282)
Last modified by gel (22282)
Numerical id 9
Author gel (22282)
Entry type Definition
Classification msc 28A05
Synonym Suslin set
Related topic PavedSpace
Related topic BorelSigmaAlgebra
Related topic UniversallyMeasurable
Defines Suslin set
Defines analytic with respect to