data described by densities.

Let $\gamma:\mathbb{R}\to\mathbb{R}^+_0$ be a nonnegative integrable function with unit total weight $\int\nolimits_\mathbb{R}\gamma(x)\,\mathrm{d}x=1$. Then $\gamma$ is called *normalised regular density* on $\mathbb{R}$.

**Quantile-based definition.**

We start again by definitions of the median as a $1/2$-quantile, see [@Haldane-Biomet48] where the so-defined median is called *arithmetic median.*

The median of a normalised regular density $\gamma$ is defined as $$\begin{aligned}
\operatorname{med}(\gamma) &:=
\min \left\{\mu\in\mathbb{R}~\left|~
\int\nolimits_{-\infty}^\mu\gamma(x)\,\mathrm{d}x
\ge\frac12\right.\right\} \;,
\label{medd-lhalf}\end{aligned}$$

The median of a normalised regular density $\gamma$ is defined as $$\begin{aligned}
\operatorname{med}(\gamma) &:= \phantom{{}\cap{}}
\left\{\mu\in\mathbb{R}~\left|~
\int\nolimits_{-\infty}^\mu\gamma(x)\,\mathrm{d}x
\ge\frac12\right.\right\} 
\notag\\*&\phantom{{}:={}}
{{}\cap{}}
\left\{\mu\in\mathbb{R}~\left|~
\int\nolimits_{\mu}^{+\infty}\gamma(x)\,\mathrm{d}x
\ge\frac12\right.\right\} \;.
\label{medd-shalf}\end{aligned}$$

For regular densities as specified so far, the inequalities $\int\gamma\,\mathrm{d}x\ge1/2$ will even be fulfilled with equality. However, and can even be used verbatim for a more general class of distribution-valued densities $\gamma\in\mathcal{D}'(\mathbb{R})$ that can be decomposed into a regular (real-valued) density $\{\gamma\}$ with a countable sum of weighted delta (Dirac) peaks, thus representing the combination of a continuous distribution of values (in which each particular number has negligible weight) with a weighted discrete set as in , . In this case the inequalities $\int\gamma\,\mathrm{d}x\ge1/2$ may not be satisfied with equality if the median happens to be at a delta peak of $\gamma$.

**Continuous half-line depth.**

The characterisation 