Projection (measure theory)

In <a href="/facts/Measure_theory/yCq7zlI4">measure theory</a>, <a href="/facts/Projection_(mathematics)/QxxC9XTp">projection</a> maps often appear when working with product (Cartesian) spaces: The <a href="/facts/Product_sigma-algebra/8LJINLNc">product sigma-algebra</a> of <a href="/facts/Measurable_space/AJ98lhT4">measurable spaces</a> is defined to be the finest such that the <a href="/facts/Projection_(set_theory)/10mwpNsM">projection mappings</a> will be <a href="/facts/Measurable_function/NgIHnb1b">measurable</a>. 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 <a href="/facts/Measurable_set/yCq7zlI4">measurable set</a> is called <a href="/facts/Analytic_set/CJn9ddXT">analytic set</a> 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.
<a href="/facts/Henri_Lebesgue/ysngVGXc">Henri Lebesgue</a> 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 <a href="/facts/Plane_(geometry)/tAUtRFcn">plane</a> onto the <a href="/facts/Real_line/6eq42xX5">real line</a> is again a Borel set. The mathematician <a href="/facts/Mikhail_Yakovlevich_Suslin/mHEmzpRr">Mikhail Yakovlevich Suslin</a> found that error about ten years later, and his following research has led to <a href="/facts/Descriptive_set_theory/ek0tvUXb">descriptive set theory</a>. The fundamental mistake of Lebesgue was to think that projection commutes with decreasing intersection, while there are simple counterexamples to that.

Projection (measure theory) open-in-new

Projection (measure theory)