I am learning how to think about probability in a more measure-theoretically sound way and there’s one thing I don’t really grasp yet.
In practice and in introductory probability classes one often works with random variables directly and does not think of them as measurable mappings between probability spaces. One then (or at least we did so) calls a random variable $X$ continuous if $P(X=x)=0$ for all $x$ in our target space. Afterwards one usually defines absolute continuity of random variables.
The question I now have is the following: suppose we have a probability space $(\Omega, \mathcal{F}, P)$ where $\Omega$ is some nice metric space (say complete) and $\mathcal{F}$ is the Borel $\sigma$-algebra. Do we then have the following implication?
Suppose $X: \Omega \to \mathbb{R}$ is a random variable. Then if $X$ is continuous in the “old” sense (i.e. $P(X=x)=0$ holds) then X is $P$-a.s. Continuous in the usual sense (i.e. as a mapping from $\Omega$ to $\mathbb{R}$ we find that the set of points where $X$ fails to be continuous is a $P$ null set.
Sorry if this question is still a bit vague!
Best Answer
To augment my above example, take $X:[0,1)\rightarrow [0,1)$ as the (continuous) identity map and assume $X$ is uniformly distributed over $[0,1)$. Define $Y:[0,1)\rightarrow[0,1)$ by $$ Y(\omega) = \left\{ \begin{array}{ll} X(\omega)&\mbox{ if $X(\omega)$ is irrational} \\ 0 & \mbox{ if $X(\omega)$ is rational and $X(\omega) \neq 0$}\\ 1/2 & \mbox{ if $X(\omega)=0$} \end{array} \right.$$ Then $P[Y=X]=1$. So $Y$ is also uniformly distributed over $[0,1)$ and so its CDF function $F_Y(y)$ is continuous for all $y \in \mathbb{R}$. But the map $Y:[0,1)\rightarrow[0,1)$ is discontinuous at every point of its domain.
This is why I do not like using the phrase "$X$ is a continuous random variable." I prefer alternatives such as: