Projection (measure theory)

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]

For an example of a non-measurable set with measurable projections, consider the space ${\displaystyle X:=\{0,1\}}$ with the 𝜎-algebra ${\displaystyle {ℱ}:=\{\varnothing ,\{0\},\{1\},\{0,1\}\}}$ and the space ${\displaystyle Y:=\{0,1\}}$ with the 𝜎-algebra ${\displaystyle {𝒢}:=\{\varnothing ,\{0,1\}\}.}$ The diagonal set ${\displaystyle \{(0,0),(1,1)\}\subseteq X\times Y}$ is not measurable relatively to ${\displaystyle {ℱ}\otimes {𝒢},}$ 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 𝜎-algebra. Let ${\displaystyle {ℒ}}$ be Lebesgue 𝜎-algebra of ${\displaystyle \mathbb{ℝ}}$ and let ${\displaystyle {{ℒ}}^{\prime }}$ be the Lebesgue 𝜎-algebra of ${\displaystyle {\mathbb{ℝ}}^2.}$ For any bounded ${\displaystyle N\subseteq \mathbb{ℝ}}$ not in ${\displaystyle {ℒ}.}$ the set ${\displaystyle N\times \{0\}}$ is in ${\displaystyle {{ℒ}}^{\prime },}$ since Lebesgue measure is complete and the product set is contained in a set of measure zero.

Still one can see that ${\displaystyle {{ℒ}}^{\prime }}$ is not the product 𝜎-algebra ${\displaystyle {ℒ}\otimes {ℒ}}$ but its completion. As for such example in product 𝜎-algebra, one can take the space ${\displaystyle \{0,1{\}}^{\mathbb{ℝ}}}$ (or any product along a set with cardinality greater than continuum) with the product 𝜎-algebra ${\displaystyle {ℱ}=\underset{t\in \mathbb{ℝ}}{⨂}{{ℱ}}_t}$ where ${\displaystyle {{ℱ}}_t=\{\varnothing ,\{0\},\{1\},\{0,1\}\}}$ for every ${\displaystyle t\in \mathbb{ℝ}.}$ In fact, in this case "most" of the projected sets are not measurable, since the cardinality of ${\displaystyle {ℱ}}$ is ${\displaystyle {\mathrm{\aleph }}_0\cdot 2^{{\mathrm{\aleph }}_0}=2^{{\mathrm{\aleph }}_0},}$ whereas the cardinality of the projected sets is ${\displaystyle 2^{2^{{\mathrm{\aleph }}_0}}.}$ 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.^[2]

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

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

An important special case of this theorem is that the projection of any Borel set of ${\displaystyle {\mathbb{ℝ}}^n}$ onto ${\displaystyle {\mathbb{ℝ}}^{n-k}}$ where ${\displaystyle k<n}$ 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 ${\displaystyle \mathbb{ℝ}}$ which is a projection of some measurable set of ${\displaystyle {\mathbb{ℝ}}^2,}$ is the only sort of such example.

• Analytic set
• Descriptive set theory – Subfield of mathematical logic

• "Measurable projection theorem", PlanetMath

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