I want to define a non-negative integer-valued random number K in which the last element of the sample space (basically the range) should be reflected in the notation of the random variable. What's the most compact notation and definition for defining such random variable?
For example:
Let $K\left(u\right)$ be non-negative integer-valued random number with sample space $\Omega = {}\{0, 1, ..., u \}$.
Basically, I want to define random variable like a function. For example, $K\left(u\right)$ is a random variable with state space $= \{0, 1, 2, ..., u\}$. So, if I write $K\left(4\right)$, the reader should understand that $K\left(4\right)$ is random variable with sample space $\{0, 1, 2, 3, 4\}$. Is my approach of defining the random variable correct or is there any better way of defining it?