Let $(p_t)_{t\geq 0}$ be a convolution semigroup of probability measures on $\mathbb R^d$ defined by $$\int _{\mathbb R^d} e^{i\langle \xi ,x\rangle }\, p_t(\mathrm d x)= e^{-t\psi (\xi )}, \quad \ \in \mathbb R^d, $$ and let $\Omega $ an open subs

In this paper, we introduce and study a notion of asymptotic expansion in measure for measurable actions. This generalizes provides new perspective on the classical strong ergodicity. Moreover, obtain structure theorems asymptotically expanding actions, showing that they admit exhaustions by domains expansion. As an application, recover recent result Marrakchi, characterizing ergodicity terms l...