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 $\map \BB \R$ be the Borel $\sigma$-algebra on $\R$.

Let $\lambda$ be the Lebesgue measure on $\struct {\R, \map \BB \R}$.

We say that $X$ is singular :


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

Also see

 * Singular Random Variable is not Absolutely Continuous