Projection (measure theory)

Summary

In measure theory, projection maps often appear when working with product spaces: The product sigma-algebra of measurable spaces is defined to be the finest such that the projection mappings will be measurable. Sometimes for some reasons product spaces are equipped with sigma-algebra different than the product sigma-algebra. In these cases the projections need not be measurable at all.

The projected set of a measurable set is called analytic set and need not be a measurable set. However, in some cases, either relatively to the product sigma-algebra or relatively to some other sigma-algebra, projected set of measurable set is indeed measurable.

Henri Lebesgue himself, one of the founders of measure theory, was mistaken about that fact. In a paper from 1905 he wrote that the projection of Borel set in the plane onto the real line is again a Borel set.[1] The mathematician Mikhail Yakovlevich Suslin found that error about ten years later, and his following research has led to descriptive set theory.[2] The fundamental mistake of Lebesgue was to think that projection commutes with decreasing intersection, while there are simple counterexamples to that.[3]

Basic examplesEdit

As an example for a non-measurable projection, one can take the space   with the sigma-algebra   and the space   with the sigma-algebra  . The diagonal set   is not measurable relatively to  , although the both projections are measurable sets.

The common example for a non-measurable set which is a projection of a measurable set, is in Lebesgue sigma-algebra. Let   be Lebesgue sigma-algebra of   and let   be the Lebesgue sigma-algebra of  . For any bounded   not in  , the set   is in  , since Lebesgue measure is complete and the product set is contained in a set of zero measure.

Still one can see that   is not the product sigma-algebra   but its completion. As for such example in product sigma-algebra, one can take the space   (or any product along a set with cardinality greater than continuum) with the product sigma-algebra   where   for every  . In fact, in this case "most" of the projected sets are not measurable, since the cardinality of   is  , whereas the cardinality of the projected sets is  . There are also examples of Borel sets in the plane which their projection to the real line is not a Borel set, as Suslin showed.[2]

Measurable projection theoremEdit

The following theorem gives a sufficient condition for the projection of measurable sets to be measurable.

Let   be a measurable space and let   be a polish space where   is its Borel sigma-algebra. Then for every set in the product sigma-algebra  , the projected set onto   is a universally measurable set relatively to  .[4]

An important special case of this theorem is that the projection of any Borel set of   onto   where   is Lebesgue-measurable, even though it is not necessarily a Borel set. In addition, it means that the former example of non-Lebesgue-measurable set of   which is a projection of some measurable set of  , is the only sort of such example.

See alsoEdit

ReferencesEdit

  1. ^ Lebesgue, H. (1905) Sur les fonctions représentables analytiquement. Journal de Mathématiques Pures et Appliquées. Vol. 1, 139–216.
  2. ^ a b Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. p. 2. ISBN 0-444-70199-0.
  3. ^ Lowther, George (8 November 2016). "Measurable Projection and the Debut Theorem". Almost Sure. Retrieved 21 March 2018.
  4. ^ * Crauel, Hans (2003). Random Probability Measures on Polish Spaces. STOCHASTICS MONOGRAPHS. London: CRC Press. p. 13. ISBN 0415273870.

External linksEdit