the subsequent section), the continuous median filter discussed in this section will strictly follow the sliding-window idea by deriving all $S(\boldsymbol{x})$ from just one set $S_0$ by shifting, $S(\boldsymbol{x})=\boldsymbol{x}+S_0:=
\{\boldsymbol{x}+\boldsymbol{y}\in\mathbb{R}^2~|~\boldsymbol{y}\in S_0\}$.

Varying the size of the neighbourhood for filtering will be an important step in the theoretical analysis, thus we precise such a procedure by a further definition.

\[def-scalsel\] Let $\varSigma$ be a map that assigns to any smooth image $u$ and any positive $\varrho$ a selector $\varSigma_{u,\varrho}$ for $u$. Assume that for each $u$ and each location $\boldsymbol{x}$ and for all $\varrho<\sigma$ the inclusion $\varSigma_{u,\varrho}(\boldsymbol{x})\subset 
\varSigma_{u,\sigma}(\boldsymbol{x})$ holds, and $\bigcap\nolimits_{\varrho>0}
\varSigma_{u,\varrho(\boldsymbol{x})}
=\{\boldsymbol{x}\}$. Then $\varSigma$ is called *scaled selector for $u$.*

Our main example of a scaled selector in this section will be the family $\mathcal{D}=\{D_\varrho~|~\varrho>0\}$ of discs with radius $\varrho$ given by $D_\varrho(\boldsymbol{x})=\{\boldsymbol{y}\in\mathbb{R}^2~|~
\lVert\boldsymbol{y-x}\rVert\le\varrho\}$.

Turning to the second step of a local filter, aggregation, we

Let $\varGamma$ be a family of integrable densities $\gamma:\mathbb{R}\to\mathbb{R}^+_0$. Let $A$ be a functional that maps densities $\gamma\in\varGamma$ to values $A(\gamma)\in\mathbb{R}$. Then $A$ is called a *$\varGamma$-aggregator on $\mathbb{R}$.*

In particular, the space-continuous median is an aggregator, where $\varGamma$ can be either the regular densities mainly considered in Section \[ssec-medd\] or the distributions $\mathcal{D}'(\mathbb{R})$. Both 