HOME

TheInfoList



OR:

In
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...
, a pushforward measure (also known as push forward, push-forward or image measure) is obtained by transferring ("pushing forward") a
measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...
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 the ...
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 in di ...
.


Definition

Given measurable spaces (X_1,\Sigma_1) and (X_2,\Sigma_2), a measurable mapping f\colon X_1\to X_2 and a measure \mu\colon\Sigma_1\to ,+\infty/math>, the pushforward of \mu is defined to be the measure f_(\mu)\colon\Sigma_2\to ,+\infty/math> given by :f_ (\mu) (B) = \mu \left( f^ (B) \right) for B \in \Sigma_. This definition applies ''
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 In mathematics, specifically measure theory, a complex measure generalizes the concept of measure by letting it have complex values. In other words, one allows for sets whose size (length, area, volume) is a complex number. Definition Formal ...
. 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 wit ...
" 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 Eucl ...
S1 (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 the ...
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 (geometry), 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 ...
R. Let ''λ'' also denote the restriction of Lebesgue measure to the interval [0, 2''π'') and let ''f'' : [0, 2''π'') → S1 be the natural bijection defined by ''f''(''t'') = exp(''i'' ''t''). The natural "Lebesgue measure" on S1 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 S1 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 tou ...
T''n''. The previous example is a special case, since S1 = T1. This Lebesgue measure on T''n'' is, up to normalization, the
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, though ...
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 Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
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 additio ...
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 nam ...
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 vector ...
''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 , the s ...
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 cons ...
to ''X'' is a Gaussian measure on R. * Consider a measurable function ''f'' : ''X'' → ''X'' and the
composition Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography *Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
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 is ...
forms a
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
. 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 In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping, ...
, i.e one for which ''f''(''μ'') = ''μ''. * One can also consider
quasi-invariant measure In mathematics, a quasi-invariant measure ''μ'' with respect to a transformation ''T'', from a measure space ''X'' to itself, is a measure which, roughly speaking, is multiplied by a numerical function of ''T''. An important class of examples ...
s 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 Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry * Equivalence class (music) *'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *''Equiva ...
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 norm ...
, 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 in di ...
can be pushed forward, the push-forward then becomes a
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
, known as the
transfer operator Transfer may refer to: Arts and media * ''Transfer'' (2010 film), a German science-fiction movie directed by Damir Lukacevic and starring Zana Marjanović * ''Transfer'' (1966 film), a short film * ''Transfer'' (journal), in management studies ...
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 In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: i ...
; as an operator on spaces of functions on measurable spaces, it is the
composition 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 ...
or Koopman operator.


See also

*
Measure-preserving dynamical system In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special cas ...


Notes


References

* * {{DEFAULTSORT:Pushforward Measure Measures (measure theory)