Definition:Random Variable/Continuous/Absolutely Continuous

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

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

Let $P_X$ be the probability distribution of $X$.

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

We say that $X$ is an absolutely continuous random variable :


 * $P_X$ is absolutely continuous with respect to $\lambda$.

Also known as
Often, particularly in elementary discussions of probability theory, the term continuous random variable is used to mean absolutely continuous random variable as defined here.

On we seek to avoid this ambiguity, and it should be made explicit whether a result applies only to absolutely continuous random variables, or general continuous random variables.

Also see

 * Absolutely Continuous Random Variable is Continuous