Koopman operator ($K_F$)

For a measurable dynamical map $F$, the Koopman operator acts on observables by $K_{F}g=g\circ F$. If $F$ is nonsingular, it defines an operator on equivalence classes of measurable functions.

For a nonsingular map $F$ on a probability space, $K_F$ extends boundedly to $L^2$ exactly when $d(F_{\ast }\mu )/d\mu \in L^{\infty }$. In that case