"Absolutely continuous r.v." vs. "continuous r.v."

I've recently come across the term "absolutely continuous random variable" in a book on measure theoretic probability. How am I supposed to distinguish between AC random variables and just continuous random variables?
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 Recognitions: Science Advisor Maybe if you consider X as a function (with random values) into some measure space, then X is AC as a function?
 Mentor Blog Entries: 8 A random variable X is called absolutely continuous if there exists a measurable function f≥0 such that [tex]P\{a

"Absolutely continuous r.v." vs. "continuous r.v."

I believe the Dirac delta distribution is an example of a distribution which is considered continuous but not absolutely continuous.

Recognitions:
 Quote by micromass A random variable X is called absolutely continuous if there exists a measurable function f≥0 such that [tex]P\{a
Right, but doesn't it come down to the same thing as f being AC as a function?

Mentor
 Quote by SW VandeCarr I believe the Dirac delta distribution is an example of a distribution which is considered continuous but not absolutely continuous.
That's too loose.

 Quote by Bacle2 Right, but doesn't it come down to the same thing as f being AC as a function?
And that's too strict.

Another way to look at a continuous random variable is that such a random variable must have P(X=x) for all x. Yet another way to look at it is that the continuous random variable has a continuous CDF. A random variable with a Dirac delta distribution violates both.

A random variable is absolutely continuous if the CDF has a derivative, call it f(x), except over a space of measure zero. There's nothing saying this function f(x) has to be continuous.

A random variable that is continuous but not absolutely continuous is called a singular random variable. One example of such a random variable would be one whose CDF is everywhere continuous but nowhere differentiable. The CDF doesn't have to be nowhere differentiable to qualify as singular. It just has to be non-differentiable over a space with a non-zero measure.

 Quote by D H Another way to look at a continuous random variable is that such a random variable must have P(X=x) for all x. Yet another way to look at it is that the continuous random variable has a continuous CDF. A random variable with a Dirac delta distribution violates both.