Let us consider that we are given a real-valued random variable $X:\Omega\to\mathbb{R}$ with a continuous probability density function under a given probability measure. I am looking for references that would tell me how to construct a discrete random variable $Y$ such that the cumulative density functions of $Y$ and $X$ are close. One problem seems to be how to select the set of values that $Y$ can take, and an additional problem seems to be how to select the probabilities over these points.
Another question that bothers me is: how to define closeness between $X$ and $Y$ (its discrete counterpart)?
I am thinking that there must be some standard text (possibly even undergraduate level) that deals with these questions, but have found none yet. I would be really grateful if you could point to an authoritative reference that covers this topic.