Distributions §

The first three classes form a partition by the Lebesgue decomposition theorem: every finite Borel measure on $\mathbb{R}^d$ decomposes uniquely as $\mu = \mu_{\text{at}} + \mu_{\text{ac}} + \mu_{\text{sg}}$ where $\mu_{\text{at}}$ is atomic (countable sum of point masses), $\mu_{\text{ac}}$ is absolutely continuous (has a density), and $\mu_{\text{sg}}$ is singular (continuous but supported on a null set, e.g. the Cantor measure). A distribution belongs to exactly one of the first three classes, or is a mixture of two or more.

The fourth class is orthogonal to the first three: it studies the algebraic and structural relationships between distributions—parametric families (exponential, location-scale, stable), closure properties (convolution, mixture, transformation), and order statistics. A given distribution (e.g. the normal) is absolutely continuous (class 2) and belongs to the exponential family (class 4); the two classifications answer different questions.

The CDF $F(x) = P(X \leq x)$ is the universal representation: it exists for every distribution regardless of type. For atomic distributions it is a step function; for absolutely continuous distributions it is differentiable with $F'(x) = f(x)$; for singular distributions it is continuous but flat almost everywhere (e.g. the Cantor function). This makes the CDF the natural bridge to Random variables and expectation, where moments, MGFs, and concentration inequalities are computed from the distribution.