The
Cantor–Bernstein–Schroeder theorem of
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
has a counterpart for
measurable space
In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured.
It captures and generalises intuitive notions such as length, area, an ...
s, sometimes called the Borel Schroeder–Bernstein theorem, since measurable spaces are also called
Borel spaces. This theorem, whose proof is quite easy, is instrumental when proving that two measurable spaces are isomorphic. The general theory of
standard Borel spaces contains very strong results about isomorphic measurable spaces, see
Kuratowski's theorem
In graph theory, Kuratowski's theorem is a mathematical forbidden graph characterization of planar graphs, named after Kazimierz Kuratowski. It states that a finite graph is planar if and only if it does not contain a Glossary of graph theory#Su ...
. However, (a) the latter theorem is very difficult to prove, (b) the former theorem is satisfactory in many important cases (see Examples), and (c) the former theorem is used in the proof of the latter theorem.
The theorem
Let
and
be measurable spaces. If there exist injective, bimeasurable maps
then
and
are isomorphic (the
Schröder–Bernstein property).
Comments
The phrase "
is bimeasurable" means that, first,
is
measurable
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts hav ...
(that is, the
preimage
In mathematics, for a function f: X \to Y, the image of an input value x is the single output value produced by f when passed x. The preimage of an output value y is the set of input values that produce y.
More generally, evaluating f at each ...
is measurable for every measurable
), and second, the
image
An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
is measurable for every measurable
. (Thus,
must be a measurable subset of
not necessarily the whole
)
An isomorphism (between two measurable spaces) is, by definition, a bimeasurable
bijection
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
. If it exists, these measurable spaces are called isomorphic.
Proof
First, one constructs a bijection
out of
and
exactly as in the
proof of the Cantor–Bernstein–Schroeder theorem. Second,
is measurable, since it coincides with
on a measurable set and with
on its complement. Similarly,
is measurable.
Examples

Example 1
The
open interval
In mathematics, a real interval is the set (mathematics), set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without ...
(0, 1) and the
closed interval
In mathematics, a real interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. A real in ...
, 1are evidently non-isomorphic as
topological spaces
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...
(that is, not
homeomorphic
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
). However, they are isomorphic as measurable spaces. Indeed, the closed interval is evidently isomorphic to a shorter closed subinterval of the open interval. Also the open interval is evidently isomorphic to a part of the closed interval (just itself, for instance).
Example 2
The real line
and the plane
are isomorphic as measurable spaces. It is immediate to embed
into
The converse, embedding of
into
(as measurable spaces, of course, not as topological spaces) can be made by a well-known trick with interspersed digits; for example,
:''g''(π,100e) = ''g''(, ) = . ….
The map
is clearly injective. It is easy to check that it is bimeasurable. (However, it is not bijective; for example, the number
is not of the form
).
References
* S.M. Srivastava, ''A Course on Borel Sets'', Springer, 1998.
:: See Proposition 3.3.6 (on page 96), and the first paragraph of Section 3.3 (on page 94).
{{DEFAULTSORT:Schroder-Bernstein theorem for measurable spaces
Theorems in measure theory
Descriptive set theory
Theorems in the foundations of mathematics