mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 i ...

''X'' to itself, is 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 ...

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 ma ...

''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 two m ...

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 wit ...

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

. Then the effect of ''T'' on μ is locally expressible as multiplication by the Jacobian determinant of the derivative (

pushforward The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things. * Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" op ...

) of ''T''. To express this idea more formally 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 ...

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 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 ...

(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 oper ...

): :

\mu' = T_ (\mu) \approx \mu.

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 s ...

. 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 ...

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 vector ...

and ''μ'' is a locally finite

Borel measure In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. F ...

on ''E'' that is quasi-invariant under all translations by elements of ''E'', then either dim(''E'') < +∞ or ''μ'' is the

trivial measure In mathematics, specifically in measure theory, the trivial measure on any measurable space (''X'', Σ) is the measure ''μ'' which assigns zero measure to every measurable set: ''μ''(''A'') = 0 for all ''A'' in Σ. Properties of the trivial mea ...

''μ'' ≡ 0.

References

{{DEFAULTSORT:Quasi-Invariant Measure Measures (measure theory) Dynamical systems

See also

References