Projection (measure theory) explained

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 -algebra different than the product -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 -algebra or relatively to some other -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

For an example of a non-measurable set with measurable projections, consider the space

X : = \{ 0 , 1 \} with the -algebra l{F} : = \{\varnothing, \{ 0 \}, \{ 1 \}, \{ 0 , 1 \}\} and the space Y : = \{ 0 , 1 \} with the -algebra l{G} : = \{\varnothing, \{ 0 , 1 \}\}. The diagonal set \{( 0 , 0 ), ( 1 , 1 )\} \subseteq X x Y is not measurable relatively to l{F} ⊗ l{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 -algebra. Let

l{L} be Lebesgue -algebra of \Reals and let l{L}' be the Lebesgue -algebra of \Reals ^{2.} For any bounded N \subseteq \Reals not in l{L}. the set N x \{ 0 \} is in l{L}', since Lebesgue measure is complete and the product set is contained in a set of measure zero.

Still one can see that

l{L}' is not the product -algebra l{L} ⊗ l{L} but its completion. As for such example in product -algebra, one can take the space \{ 0 , 1 \} ^{\Reals} (or any product along a set with cardinality greater than continuum) with the product -algebra l{F} = style { o times\limits _{t\in\Reals} } \mathcal_t where l{F} _{t} = \{\varnothing,\{ 0 \} ,\{ 1 \} ,\{ 0 , 1 \}\} for every t \in \Reals. In fact, in this case "most" of the projected sets are not measurable, since the cardinality of l{F} is \aleph _{0} ⋅ 2 ^{\aleph _{0}} = 2 ^{\aleph _{0}} , whereas the cardinality of the projected sets is 2 ^{2 ^{\aleph _{0}}} . 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.

Measurable projection theorem

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

Let

(X, l{F}) be a measurable space and let (Y, l{B}) be a polish space where l{B} is its Borel -algebra. Then for every set in the product -algebra l{F} ⊗ l{B}, the projected set onto X is a universally measurable set relatively to l{F}. ^[4]

An important special case of this theorem is that the projection of any Borel set of

\Reals ^{n} onto \Reals ^{n-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 \Reals which is a projection of some measurable set of \Reals ^{2,} is the only sort of such example.

External links

• "Measurable projection theorem", PlanetMath

Notes and References

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