In
mathematics, a quasi-invariant measure ''μ'' with respect to a transformation ''T'', from a
measure space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
''X'' to itself, is a
measure which, roughly speaking, is multiplied by a
numerical function of ''T''. An important class of examples occurs when ''X'' is a
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One m ...
''M'', ''T'' is a
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given tw ...
of ''M'', and ''μ'' is any measure that locally is a
measure with base the
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
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
. Then the effect of ''T'' on μ is locally expressible as multiplication by the
Jacobian
In mathematics, a Jacobian, named for Carl Gustav Jacob Jacobi, may refer to:
*Jacobian matrix and determinant
*Jacobian elliptic functions
*Jacobian variety
*Intermediate Jacobian
In mathematics, the intermediate Jacobian of a compact Kähler m ...
determinant of the derivative (
pushforward) of ''T''.
To express this idea more formally in
measure theory terms, the idea is that the
Radon–Nikodym derivative of the transformed measure μ′ with respect to ''μ'' should exist everywhere; or that the two measures should be
equivalent (i.e. mutually
absolutely continuous
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central ope ...
):
:
That means, in other words, that ''T'' preserves the concept of a set of
measure zero
In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length.
The notion of null ...
. Considering the whole equivalence class of measures ''ν'', equivalent to ''μ'', it is also the same to say that ''T'' preserves the class as a whole, mapping any such measure to another such. Therefore, the concept of quasi-invariant measure is the same as ''invariant measure class''.
In general, the 'freedom' of moving within a measure class by multiplication gives rise to
cocycles, when transformations are composed.
As an example,
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 ...
on
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
R
''n'' is not invariant under translation (like Lebesgue measure is), but is quasi-invariant under all translations.
It can be shown that if ''E'' is 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 ...
and ''μ'' is a
locally finite Borel measure on ''E'' that is quasi-invariant under all translations by elements of ''E'', then either dim(''E'') < +∞ or ''μ'' is the
trivial measure ''μ'' ≡ 0.
See also
*
Invariant measure
References
{{DEFAULTSORT:Quasi-Invariant Measure
Measures (measure theory)
Dynamical systems