This article does not cite any sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: "Quasi-invariant measure" – news · newspapers · books · scholar · JSTOR (December 2009) (Learn how and when to remove this message)

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 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):

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.