Definition:Random Variable/Continuous/Singular

Definition
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be a continuous random variable on $\struct {\Omega, \Sigma, \Pr}$.

Let $\lambda$ be the Lebesgue measure on $\R$.

We say that $X$ is singular :


 * there exists a $\lambda$-null set $B \subseteq \R$ such that $\map \Pr {X \in B} = 0$.

Also see

 * Singular Random Variable is not Absolutely Continuous