Projection (measure theory)

In measure theory, projection maps  often appear when working with product (Cartesian) 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 𝜎-algebra different than the product 𝜎-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 𝜎-algebra or relatively to some other 𝜎-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]

## Contents

* 1 Basic examples
* 2 Measurable projection theorem
* 3 See also
* 4 References
* 5 External links

## Basic examples

For an example of a non-measurable set with measurable projections, consider the space X:={0,1} with the 𝜎-algebra ℱ:={∅,{0},{1},{0,1}} and the space Y:={0,1} with the 𝜎-algebra 𝒢:={∅,{0,1}}. The diagonal set {(0,0),(1,1)}⊆X×Y is not measurable relatively to ℱ⊗𝒢, 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 𝜎-algebra. Let ℒ be Lebesgue 𝜎-algebra of ℝ and let ℒ′ be the Lebesgue 𝜎-algebra of ℝ2. For any bounded N⊆ℝ not in ℒ. the set N×{0} is in ℒ′, since Lebesgue measure is complete and the product set is contained in a set of measure zero.

Still one can see that ℒ′ is not the product 𝜎-algebra ℒ⊗ℒ but its completion. As for such example in product 𝜎-algebra, one can take the space {0,1}ℝ (or any product along a set with cardinality greater than continuum) with the product 𝜎-algebra ℱ=⨂t∈ℝℱt where ℱt={∅,{0},{1},{0,1}} for every t∈ℝ. In fact, in this case "most" of the projected sets are not measurable, since the cardinality of ℱ is ℵ0⋅2ℵ0=2ℵ0, whereas the cardinality of the projected sets is 22ℵ0. 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 (X,ℱ) be a measurable space and let (Y,ℬ) be a polish space where ℬ is its Borel 𝜎-algebra. Then for every set in the product 𝜎-algebra ℱ⊗ℬ, the projected set onto X is a universally measurable set relatively to ℱ.[4]

An important special case of this theorem is that the projection of any Borel set of ℝn onto ℝn−k where k<n 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 ℝ which is a projection of some measurable set of ℝ2, is the only sort of such example.

## See also

* Analytic set
* Descriptive set theory - Subfield of mathematical logic

## References

1. ↑ Lebesgue, H. (1905) Sur les fonctions représentables analytiquement. Journal de Mathématiques Pures et Appliquées. Vol. 1, 139–216.
2. ↑ 2.0 2.1 Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. p. 2. ISBN 0-444-70199-0. https://www.math.ucla.edu/~ynm/books.htm.
3. ↑ Lowther, George (8 November 2016). "Measurable Projection and the Debut Theorem". https://almostsure.wordpress.com/2016/11/08/measurable-projection-and-the-debut-theorem/#more-1873.
4. ↑ * Crauel, Hans (2003). Random Probability Measures on Polish Spaces. STOCHASTICS MONOGRAPHS. London: CRC Press. p. 13. ISBN 0415273870. https://books.google.com/books?id=RhT_aZGqmQEC&dq=measurable+projection+theorem&pg=PA13.

## External links

* "Measurable projection theorem", PlanetMath

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