Projection (measure theory)
In measure theory, projection map plays an important role in treating 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 examples
As an example for not 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 theorem
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 in universally measurable set relatively to .[4]
An important spacial case of this theorem is that the projection of any Borel set ot 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 also
External links
References
- ↑ Lebesgue, H. (1905) Sur les fonctions représentables analytiquement. Journal de Mathématiques Pures et Appliquées. Vol. 1, 139–216.
- 1 2 Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. p. 2. ISBN 0-444-70199-0.
- ↑ Lowther, George (8 November 2016). "Measurable Projection and the Debut Theorem". Almost Sure. Retrieved 21 March 2018.
- ↑
- Crauel, Hans (2003). Random Probability Measures on Polish Spaces. STOCHASTICS MONOGRAPHS. London: CRC Press. p. 13. ISBN 0415273870.