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The name is a back-formation from the Dirac delta function, considered as a Schwartz distribution, for example on the real line; measures can be taken to be a special kind of distribution.
The simplest example of a discrete measure on the real line is the Dirac delta function
For example, while a general quasisymmetric map in the complex plane could map the real line to a set of Hausdorff dimension strictly greater than one, a δ-monotone will always map the real line to a rotated graph of a Lipschitz function L:ℝ → ℝ.
where R is the real line, is a (positive definite) metric and is a positive function on the Riemannian manifold S.