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):

:$mu\text{'}\; =\; T\_\{*\}\; (mu)\; approx\; mu.$

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.