leaves unchanged, a so-called invariant measure, i.e one for which f∗(μ) = μ. 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. Random variables induce pushforward measures. They map a probability space into a codomain space and endow that space with a probability measure defined by the pushforward. Furthermore, because random variables are functions (and hence total functions), the inverse image of the whole codomain is the whole domain, and the measure of the whole domain is 1, so the measure of the whole codomain is 1. This means that random variables can be composed ad infinitum and they will always remain as random variables and endow the codomain spaces with probability measures. A generalization In general, any measurable function can be pushed forward, the push-forward then becomes a linear operator, known as the transfer operator or Frobenius–Perron operator. In finite spaces this operator typically satisfies the requirements of the Frobenius–Perron theorem, and the maximal eigenvalue of the operator corresponds to the invariant measure. The adjoint to the push-forward is the pullback; as an operator on spaces of functions on measurable spaces, it is the composition operator or Koopman operator. See also Measure-preserving dynamical system Notes References Measures (measure theory)