Definition 2

A family $(\mu_\alpha)_{\alpha \in I}$, where, for every $\alpha \in I$, $\mu_\alpha$ is a measure on $X_\alpha$, is said to be an inverse system of measures if, whenever $\alpha \leq \beta$, $\mu_\alpha = (p_{\alpha\beta})_*(\mu_\beta)$. A measure $\mu$ on $X = \lim_{\leftarrow} X_\alpha$ is said to be an inverse limit of the inverse system $(\mu_\alpha)$ if $\mu_\alpha = (p_\alpha)_*(\mu)$ for every $\alpha \in I$.