Projection (measure theory)

Summary

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]

Basic examples

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For an example of a non-measurable set with measurable projections, consider the space   with the 𝜎-algebra   and the space   with the 𝜎-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 𝜎-algebra. Let   be Lebesgue 𝜎-algebra of   and let   be the Lebesgue 𝜎-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 measure zero.

Still one can see that   is not the product 𝜎-algebra   but its completion. As for such example in product 𝜎-algebra, one can take the space   (or any product along a set with cardinality greater than continuum) with the product 𝜎-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

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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 𝜎-algebra. Then for every set in the product 𝜎-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 also

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  • Analytic set – subset of a Polish space that is the continuous image of a Polish space
  • Descriptive set theory – Subfield of mathematical logic

References

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  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.
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