In
measure theory, a pushforward measure (also known as push forward, push-forward or image measure) is obtained by transferring ("pushing forward") a
measure from one
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 ...
to another using 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 i ...
.
Definition
Given
measurable spaces and
, a measurable mapping
and a measure
mutatis mutandis
''Mutatis mutandis'' is a Medieval Latin phrase meaning "with things changed that should be changed" or "once the necessary changes have been made". It remains unnaturalized in English and is therefore usually italicized in writing. It is used i ...
'' for a
signed or
complex measure.
The pushforward measure is also denoted as
\mu \circ f^,
f_\sharp \mu,
f \sharp \mu, or
f \# \mu.
Main property: change-of-variables formula
Theorem:
[Sections 3.6–3.7 in ] A measurable function ''g'' on ''X''
2 is integrable with respect to the pushforward measure ''f''
∗(''μ'') if and only if the composition
g \circ f is integrable with respect to the measure ''μ''. In that case, the integrals coincide, i.e.,
:
\int_ g \, d(f_* \mu) = \int_ g \circ f \, d\mu.
Note that in the previous formula
X_1=f^(X_2).
Examples and applications
* A natural "
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides ...
" on the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
S
1 (here thought of as a subset of the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...
C) may be defined using a push-forward construction and Lebesgue measure ''λ'' on 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 po ...
R. Let ''λ'' also denote the restriction of Lebesgue measure to the interval [0, 2''π'') and let ''f'' : [0, 2''π'') → S
1 be the natural bijection defined by ''f''(''t'') = exp(''i'' ''t''). The natural "Lebesgue measure" on S
1 is then the push-forward measure ''f''
∗(''λ''). The measure ''f''
∗(''λ'') might also be called "arc length measure" or "angle measure", since the ''f''
∗(''λ'')-measure of an arc in S
1 is precisely its arc length (or, equivalently, the angle that it subtends at the centre of the circle.)
* The previous example extends nicely to give a natural "Lebesgue measure" on the ''n''-dimensional
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not ...
T
''n''. The previous example is a special case, since S
1 = T
1. This Lebesgue measure on T
''n'' is, up to normalization, the
Haar measure for the
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
,
connected Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...
T
''n''.
*
Gaussian measure
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space R''n'', closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are named ...
s on infinite-dimensional vector spaces are defined using the push-forward and the standard Gaussian measure on the real line: a
Borel measure ''γ'' on a
separable Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between ve ...
''X'' is called Gaussian if the push-forward of ''γ'' by any non-zero
linear functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , th ...
in the
continuous dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
to ''X'' is a Gaussian measure on R.
* Consider a measurable function ''f'' : ''X'' → ''X'' and the
composition of ''f'' with itself ''n'' times:
::
f^ = \underbrace_ : X \to X.
: This
iterated function
In mathematics, an iterated function is a function (that is, a function from some set to itself) which is obtained by composing another function with itself a certain number of times. The process of repeatedly applying the same function ...
forms 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 i ...
. It is often of interest in the study of such systems to find a measure ''μ'' on ''X'' that the map ''f'' leaves unchanged, a so-called
invariant measure, i.e one for which ''f''
∗(''μ'') = ''μ''.
* One can also consider
quasi-invariant measures for such a dynamical system: a measure ''
\mu'' on ''
(X,\Sigma)'' is called quasi-invariant under
f if the push-forward of ''
\mu'' by
f is merely
equivalent to the original measure ''μ'', not necessarily equal to it. A pair of measures
\mu, \nu on the same space are equivalent if and only if
\forall A\in \Sigma: \ \mu(A) = 0 \iff \nu(A) = 0, so
\mu is quasi-invariant under
f if
\forall A \in \Sigma: \ \mu(A) = 0 \iff f_* \mu(A) = \mu\big(f^(A)\big) = 0
* Many natural probability distributions, such as the
chi distribution
In probability theory and statistics, the chi distribution is a continuous probability distribution. It is the distribution of the positive square root of the sum of squares of a set of independent random variables each following a standard no ...
, can be obtained via this construction.
* Random variables induce pushforward measures. They map a probability space into a codomain space and endow that space with a probability measure defined by the pushforward. Furthermore, because random variables are functions (and hence total functions), the inverse image of the whole codomain is the whole domain, and the measure of the whole domain is 1, so the measure of the whole codomain is 1. This means that random variables can be composed ''ad infinitum'' and they will always remain as random variables and endow the codomain spaces with probability measures.
A generalization
In general, any
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 i ...
can be pushed forward, the push-forward then becomes a
linear operator, known as the
transfer operator or
Frobenius–Perron operator. In finite spaces this operator typically satisfies the requirements of the
Frobenius–Perron theorem, and the maximal eigenvalue of the operator corresponds to the invariant measure.
The adjoint to the push-forward is the
pullback; as an operator on spaces of functions on measurable spaces, it is the
composition operator or
Koopman operator
In mathematics, the composition operator C_\phi with symbol \phi is a linear operator defined by the rule
C_\phi (f) = f \circ \phi
where f \circ \phi denotes function composition.
The study of composition operators is covered bAMS category 47B33 ...
.
See also
*
Measure-preserving dynamical system
Notes
References
*
*
{{DEFAULTSORT:Pushforward Measure
Measures (measure theory)