In
mathematics, an invariant measure is a
measure that is preserved by some
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
. The function may be a
geometric transformation. For examples,
circular angle is invariant under rotation,
hyperbolic angle is invariant under
squeeze mapping
In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is ''not'' a rotation or shear mapping.
For a fixed positive real number , th ...
, and a difference of
slopes is invariant under
shear mapping.
Ergodic theory is the study of invariant measures in
dynamical systems. The
Krylov–Bogolyubov theorem proves the existence of invariant measures under certain conditions on the function and space under consideration.
Definition
Let
be a
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.
Definition
Consider a set X and a σ-algebra \mathcal A on X. Then t ...
and let
be a
measurable function
In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in d ...
from
to itself. A measure
on
is said to be invariant under
if, for every measurable set
in
In terms of the
pushforward measure, this states that
The collection of measures (usually
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more ge ...
s) on
that are invariant under
is sometimes denoted
The collection of
ergodic measures,
is a subset of
Moreover, any
convex combination
In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1. In other wor ...
of two invariant measures is also invariant, so
is a
convex set;
consists precisely of the extreme points of
In the case of a
dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
where
is a measurable space as before,
is a
monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids a ...
and
is the flow map, a measure
on
is said to be an invariant measure if it is an invariant measure for each map
Explicitly,
is invariant
if and only if
Put another way,
is an invariant measure for a sequence of
random variables
(perhaps a
Markov chain or the solution to a
stochastic differential equation) if, whenever the initial condition
is distributed according to
so is
for any later time
When the dynamical system can be described by a
transfer operator, then the invariant measure is an eigenvector of the operator, corresponding to an eigenvalue of
this being the largest eigenvalue as given by the
Frobenius-Perron theorem.
Examples
* Consider the
real line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
with its usual
Borel σ-algebra
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named ...
; fix
and consider the translation map
given by:
Then one-dimensional
Lebesgue measure is an invariant measure for
* More generally, on
-dimensional
Euclidean space with its usual Borel σ-algebra,
-dimensional Lebesgue measure
is an invariant measure for any
isometry of Euclidean space, that is, a map
that can be written as
for some
orthogonal matrix
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.
One way to express this is
Q^\mathrm Q = Q Q^\mathrm = I,
where is the transpose of and is the identity ma ...
and a vector
* The invariant measure in the first example is unique up to trivial renormalization with a constant factor. This does not have to be necessarily the case: Consider a set consisting of just two points
and the identity map
which leaves each point fixed. Then any probability measure
is invariant. Note that
trivially has a decomposition into
-invariant components
and
*
Area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an open s ...
measure in the Euclidean plane is invariant under the
special linear group of the
real matrices of
determinant
* Every
locally compact group has a
Haar measure that is invariant under the group action.
See also
*
References
* John von Neumann (1999) ''Invariant measures'',
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
{{DEFAULTSORT:Invariant Measure
Dynamical systems
Measures (measure theory)