Theorem (Lebesgue decomposition): Any probability distribution $F$ can uniquely be decomposed as

\[F = F_\text{D} + F_\text{AC} + F_\text{SC},\]

where

• $F_\text{D}$ is the discrete component (i.e., probability is given by a sum of point masses)
• $F_\text{AC}$ is the absolutely continuous component (i.e., probability is given by an integral with respect to a density function)
• $F_\text{AC}$ is the singular continuous component (i.e, it’s neither of the above)