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.
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] For an example of a non-measurable set with measurable projections, consider the space
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.
be Lebesgue 𝜎-algebra of
since Lebesgue measure is complete and the product set is contained in a set of measure zero.
is not the product 𝜎-algebra
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
In fact, in this case "most" of the projected sets are not measurable, since the cardinality of
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.
is its Borel 𝜎-algebra.
Then for every set in the product 𝜎-algebra
is a universally measurable set relatively to
[4] An important special case of this theorem is that the projection of any Borel set of
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.