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 by ''T''. An important class of examples occurs when ''X'' is a smooth manifold ''M'', ''T'' is a diffeomorphism of ''M'', and ''μ'' is any measure that locally is a measure with base the Lebesgue measure on Euclidean space. Then the effect of ''T'' on μ is locally expressible as multiplication by the Jacobian 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):
:$mu\text{'}\; =\; T\_\{$
★ } (mu) pprox mu.
That means, in other words, that ''T'' preserves the concept of a set of measure zero. 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 on Euclidean space '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 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.