User:NikelsenH/Asymptotic invariance

Definition
Let $$ I $$ be either $$ \mathbb R $$ or $$ \N $$ and let $$ \|\mu\|_{tv} $$ be the total variation norm of the measure $$ \mu $$.

Let $$ (\mu_i)_{i \in I} $$ be a series of measures on $$ \R^n $$ that satisfy
 * $$ \sup_{i \in I} \{ \; \|\mu\|_{tv} \mid i \in I\} < \infty $$.

Then the series $$ (\mu_i)_{i \in I} $$ is called asymptotically invariant if
 * $$ \|\mu_i-\mu_i*\delta_x\|_{tv} \to 0 \text{ as } i \to \infty $$

for all $$ x \in \R^n $$. Here $$ * $$ denotes the convolution and $$ \delta_x $$ the Dirac measure at the point $$ x $$.