Remark

The giving of a measure $\mu$ on a locally compact space X defines on X (along with the topology of X) a structure $\mathscr S$. Let $X_1$ be a second set, $\varphi$ a bijective mapping of X onto $X_1$; in conformity with general definitions (S, R, §8), the structure $\mathscr S_1$ obtained by transporting to $X_1$ the structure $\mathscr S$ of X, by means of $\varphi$, is defined in the following way. The topology of X is transported to $X_1$ by $\varphi$; the functions of $\mathscr K(X_1;\mathbf C)$ are then the functions $f$ such that $f\circ\varphi$ belongs to $\mathscr K(X;\mathbf C)$, and the measure $\mu_1$ on $X_1$ is defined by $\mu_1(f)=\mu(f\circ\varphi)$.

In particular, an automorphism of the structure $\mathscr S$ is a homeomorphism $\sigma$ of X onto itself, such that

for every function $f\in\mathscr K(X;\mathbf C)$; the measure $\mu$ is then also said to be invariant under the homeomorphism $\sigma$.