Definition of an absolutely continuous random variable

Question

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.

Answer 1

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...