
Propagation of Uncertainty in a Continuous Time Dynamical System 279
Lemma 7.4.1 Let (Ω, F, µ) be a σ-finite measure space, and η : Ω → Ω be
an invertible non-singular transformation (i.e., η
−1
non-singular) and P be
the associated Frobenius –Perron operator. Then for every f ∈ L
1
(Ω, F, µ) we
have
P(f(x)) = f (η
−1
(x))J
−1
(x), (7.132)
where J(x) is the determinant of the Jacobian matrix
∂η(x)
∂x
.
Koopman operator The Koopman operator is widely used in applications,
as it can be used to describe the evolution of the observables of a system on
the phase spac e. Its definitio n is given as follows.
Definition 7.4.5 Let (Ω, F, µ) be a σ-finite measure space. If