Lebesgue's decomposition shows that a probability distribution can be decomposed into absolute continuous, discrete, and singular-continuous parts. But in literature, different terminologies are used, which I want to unreval in what follows:
- Continuous random variables have absolute-continuous probability distributions. Their densities are probability density functions.
- Discrete random variables are described by probabilities. Their densities is termed probability mass function. Hence, the distribution is singular.
- Distributions that are both, singular and continuous, are often called singular. Singular-continuous distributions have a cumulative distribution function (devil's staircase) similar to fractals: the function is continuous, but no derivation exists. It has plateaus like that of discrete distributions, but no jumps. In dimensions greater than unity, singular-continuous distributions may be induced by dimensions of absolute-continuous and discrete nature.