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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]}

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]}

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.

**^**Lebesgue, H. (1905) Sur les fonctions représentables analytiquement. Journal de Mathématiques Pures et Appliquées. Vol. 1, 139–216.- ^
^{a}^{b}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.

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