How to define sample space for discrete random variable If $S$ is the sample space of some discrete random variable $X$, what is usually given as its superset? $S \subset \mathbb{R}$ or $S \in \mathbb{Q}$? The $X$'s I have are digitized biomedical data.
Edit 
I'm writing an introduction to biomedical data analysis. As a part of it, I must provide some introduction to statistics. The reason I came to ask this is that I don't know statistics very well myself. Anyway, I was assigned to write this up so I'm trying to do it the right way, and learn something myself too. 
So, like I wrote the situation is that I have some digitized biomedical data. There was some continuous signal that was sampled (thus its discrete in time) and then quantized (after which the signal may have only certain discrete values).
My sample space is the set of values the signal may take. Usually one sample is presented with 16 bits, so there's 65536 possible values. However, the sample values are not presented as integers but decimal values like 0.101 or 0.521.
The confusion came from that in principal there the one-to-one mapping like you have written but clearly 0.101 is not an integer.
I don't have very strong mathematical / statistical background, so I'm pretty much learning the basics still (for the data I have I am just the "end user").
 A: If it is actually discrete, then it should be possible to place your sample space in one-to-one correspondence with (some subset of) $\mathbb{Z}$. That's more or less the definition of discrete. 
However, the actual values a discrete random variable takes can come from anywhere. The sample space for the roll of a dice is $S = \{1,2,3,4,5,6\}$ and the individual values are obviously integers too. On the other hand, suppose you bin circular data into four quadrants. The centers of your bins might be at $S = \{45^{\circ}, 135^{\circ}, 225^{\circ}, 315^{\circ}\}$, which are also in $\mathbb{Z}$. If you convert them to radians, then $S = \{ \frac{\pi}{4}, \frac{3\pi}{4},  \frac{5\pi}{4} \frac{7\pi}{4} \}$, which are clearly in $\mathbb{R}$, but not $\mathbb{Z}$ or $\mathbb{Q}$, but it is pretty easy to map them back onto $\mathbb{Z}$--just divide by $\frac{\pi}{4}$.
I can't think of a situation where you would care too much about what set $S$ was a subset of, but the correspondence might be useful if you need to show that something is discrete. 
