Definition of an absolutely continuous random variable Just what is the proper definition of an absolutely continuous random variable? It's supposed to be something like:
$$\mathbf{P} (A) = \int_A f d \mu$$
for some Borel set $A$. But what is $\mu$? Is it the Lebesgue measure? The Borel measure on $\mathbb{R}$?  Sometimes the definition is also done with Riemann integrals where $A=(-\infty, t]$. My textbook is very vague on this so I'd like to know which notion is the commonly accepted one.
 A: A  real random variable $X$ defines a measure on the Borel (Lebesgue) sets of the real line:
$$Q_X(A)=P(X\in A).$$
Let $\mu$ denote the Lebesgue measure. If $Q_X$ is absolute contionuous with respect to $\mu$, that is, if $\mu(A)=0$ implies that $Q_A(A)=0$ then there exists a function $f_X$, the so called pdf. of $X$, such that $$Q_X(A)=\int_Af_X\ d\mu.$$
It follows that for the special $$A_t=\{x:x<t\}$$ one can define the so called cdf. function by $$F_X(t)=Q_X(A_t)=\int_{A_t}fd\mu.$$
The following questions arise:


*

*What properties of a cdf. guarantee that it can be calculated as a Riemann integral:
$$F_X(t)=\int_{A_t}f_Xd\mu=\int_{-\infty}^{t}f(t)dt?$$

*Does $F_X$ uniquely determine a $Q_X$?

*Does $F_X$ or $Q_X$ determine a measure space $(\Omega,\mathscr A, P)$ on which $$X: (\Omega,\mathscr A, P) \rightarrow (\mathbb R,\mathscr L ,Q_X), $$ is a measurable function?

*Is the pdf. the derivative of the cdf.?
The texts usually simplify these questions and define the cdf. as
$$F_X(t)=P(X<t)$$ and the pdf. as the derivative of the cdf.:
$$f_X(t)=\frac{dF_X(t)}{dt}$$ if this derivative exists.
These texts then treat separately the case when $X$ takes only distinct real values. Also, the average texts do not even deal with the more complicated cases.
It is very hard to write a good text book on probability...
