In
. Sometimes for some reasons product spaces are equipped with đ-algebra different than
the product đ-algebra. In these cases the projections need not be measurable at all.
The projected set of a
and need not be a measurable set. However, in some cases, either relatively to the product đ-algebra or relatively to some other đ-algebra, projected set of measurable set is indeed measurable.
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
with the đ-algebra
and the space
with the đ-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 đ-algebra
. Let
![{\displaystyle {\mathcal {L}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9027196ecb178d598958555ea01c43157d83597c)
be Lebesgue đ-algebra of
![{\displaystyle \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc)
and let
![{\displaystyle {\mathcal {L}}'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db54165ac6b5654b79fff6561808660353fffc78)
be the Lebesgue đ-algebra of
![{\displaystyle \mathbb {R} ^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/066b155c535a38739cc0c4b288324cbb7a4a227a)
For any bounded
![{\displaystyle N\subseteq \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/675445b1cb7a520889ffb944c86d099d0344cca4)
not in
![{\displaystyle {\mathcal {L}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b4b7539d7cc56c65fcf1a335544780d1ce365ed)
the set
![{\displaystyle N\times \{0\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dcec4d75561b3ad733786b839b3da2142c0631a8)
is in
![{\displaystyle {\mathcal {L}}',}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ae45e35a28c2247c5ba82637439d7bf5358b7c6)
since
Lebesgue measure is
complete and the product set is contained in a set of measure zero.
Still one can see that
is not the product đ-algebra
but its completion. As for such example in product đ-algebra, one can take the space
(or any product along a set with cardinality greater than continuum) with the product đ-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]
Measurable projection theorem
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 đ-algebra. Then for every set in the product đ-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.
See also
- Analytic set â subset of a Polish space that is the continuous image of a Polish spacePages displaying wikidata descriptions as a fallback
- Descriptive set theory â Subfield of mathematical logic
References
External links