Suppose $n\in\mathbb{N}$ and set $A\subseteq\mathbb{R}^{n}$.


If we define a sequence of sets $\left(F\_r\right)\_{r\in\mathbb{N}}$ with a [set theoretic limit](https://en.wikipedia.org/wiki/Set-theoretic_limit) of $A$; how do we define the rate *at* which $\left(F\_r\right)\_{r\in\mathbb{N}}$ converges to $A$?



Convergence rate of a sequence of sets to a set-theoretic limit?
$\newcommand\R{\mathbb R}\newcommand\si\sigma\newcommand\Si\Sigma$If $\mu$ is a finite measure on a $\si$-algebra $\Si$ over $\R^n$ and $(F\_r)$ is sequence in $\Si$ such that $\lim\_r F\_r=A$, then ($A\in\Si$ and) $\lim\_r \mu(F\_r)=\mu(A)$.
So then, one may define the rate of convergence of $(F\_r)$ to $A$ as the rate of convergence of $(\mu(F\_r))$ to $\mu(A)$.


Alternatively, again if $\mu$ is a finite measure on a $\si$-algebra $\Si$ over $\R^n$ and $(F\_r)$ is sequence in $\Si$ such that $\lim\_r F\_r=A$, then $\lim\_r (F\_r+A)=\emptyset$ and hence $\lim\_r \mu(F\_r+A)=0$, where $+$ denotes the symmetric difference.
So then, one may define the rate of convergence of $(F\_r)$ to $A$ as the rate of convergence of $(\mu(F\_r+A))$ to $0$. (Cf. e.g. [this paper](https://link.springer.com/article/10.1007/s11117-017-0507-8) or [its arXiv version](https://arxiv.org/abs/1702.01142).)


