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Are there standard definitions that express the notion that sampling from a given continuous random variable can be approximated "to any desired degree of accuracy" by sampling from an appropriately chosen discrete random variable?

In applications of sampling, people usually aren't concerned that the realized values of continuous random variables (such as voltages, masses, etc.) are only measured to a finite precision. They do estimation and hypothesis testing with the rounded-off data. To express that faith in the context of estimation my first attempt at a definition is:

Let $X$ be a real valued random variable with probability density $f(x)$ where $f$ is defined by a finite set of parameters ${p_1,p_2,.. p_m}$. To say that sampling from $X$ is approximable by discrete sampling shall mean: For each function $g(p_1,p_2,..,p_m)$ of the parameters of $f$ and for each integer $N > 0 $ and for each estimator $\hat{g}(x_1,x_2...x_N) $ of $g$ defined on $N$ independent random samples of $X$ and each symmetric $q$% confidence interval $C$ for $\hat{g}$ there exists a discrete real valued random variable $Y$ such that the estimator $\hat{g}$ evaluated on $N$ independent random samples of $Y$ has a symmetric q% confidence interval for estimating $g$ that has length equal or less than the length of $C$.

Is this a reasonable definition? Is it overly optimistic in the sense that it might make sampling the commonly used continuous probability distribution un-approximable by discrete sampling. Perhaps $g(p_1,p_2,...p_m)$ should be restricted to be a $k$_th moments of $f(x)$ and $\hat{g}$ should be restricted to a nice class of functions.

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