Pushforward measure

Definition

Main property: change-of-variables formula

Examples and applications

• A natural "Lebesgue measure" on the unit circle S¹ and let f : 0, 2π) → S¹ be the natural bijection defined by f = exp. The natural "Lebesgue measure" on S¹ is then the push-forward measure f_∗. The measure f_∗ might also be called "arc length measure" or "angle measure", since the f_∗-measure of an arc in S¹ is precisely its [arc length
• The previous example extends nicely to give a natural "Lebesgue measure" on the n-dimensional torus Tⁿ. The previous example is a special case, since S¹ = T¹. This Lebesgue measure on Tⁿ is, up to normalization, the Haar measure for the compact, connected Lie group Tⁿ.
• Gaussian measures on infinite-dimensional vector spaces are defined using the push-forward and the standard Gaussian measure on the real line: a Borel measure γ on a separable Banach space X is called Gaussian if the push-forward of γ by any non-zero linear functional in the continuous dual space to X is a Gaussian measure on R.
• Consider a measurable function f : X → X and the composition of f with itself n times:
• One can also consider quasi-invariant measures for such a dynamical system: a measure ' on ' is called quasi-invariant under if the push-forward of by is merely equivalent to the original measure μ, not necessarily equal to it. A pair of measures on the same space are equivalent if and only if, so is quasi-invariant under if
• Many natural probability distributions, such as the chi distribution, can be obtained via this construction.

  A generalization