Singular distribution
A singular distribution or singular continuous distribution is a probability distribution concentrated on a set of Lebesgue measure zero, for which the probability of each point in that set is zero.
Properties
Such distributions are not absolutely continuous with respect to Lebesgue measure.A singular distribution is not a discrete probability distribution because each discrete point has a zero probability. On the other hand, neither does it have a probability density function, since the Lebesgue integral of any such function would be zero.
In general, distributions can be described as a discrete distribution, an absolutely continuous distribution, a singular distribution, or can be decomposed into a mixture of these.