Singular distribution

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In probability, a singular distribution is a probability distribution concentrated on a set of Lebesgue measure zero, where the probability of each point in that set is zero. These distributions are sometimes called singular continuous distributions. 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.

An example is the Cantor distribution.

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