# Confusing definition of discrete variables?

I am reading "introduction to probability 2nd edition" by Joseph K. and Blitzstein. The state a definition of discrete random variables as follows:

A random variable X is said to be discrete if there is a finite list of values $$a_1, a_2, . . . , a_n$$ or an infinite list of values $$a_1, a_2, . . .$$ such that $$P(X = a_j \text{ for some j}) = 1$$.

P is the probability function. So with this definition, I am confused about what is meant by "such that $$P(X = a_j \text{ for some j}) = 1$$" because from my experience "for some j" usually means "for one specific j". But anyone can imagine an experiment with a sample space $$S$$ consisting of multiple events where no single event with have 100% chance of happening. So what is meant by this definition with respect to the probability function P?

Of course, my intuitive guess would be that they mean there must exist a certain number of events such that the accumulated probability of those is 100% - am I right and is the use of "for some" in the definition standard practice and thus there is in fact nothing wrong with it?

• I agree the phrasing isn't ideal, but they just mean $P(X\not \in \{a_i\})=0$.
– lulu
Oct 1 '19 at 14:38

Noting that any list in the form of $$a_1,a_2,\cdots$$ is countable. Let $$A=\cup_{i\in \mathbb{N}}\{a_i\}$$Then $$P(X=a_i\text{ for some i})=1 \iff P(X\in A)=1\iff P(X\notin A)=0$$ In this case, the total probability can be expressed as a sum$$1=\sum_{a\in A}P(X=a)$$ If it is impossible to give a countable set $$A$$ such that $$P(X\in A)=1$$, then we must be in the case that $$\sum_{a\in A}P(X=a)<1 \qquad \text{for any countable }A$$ So the total probability will have to be expressed as an integral (or a mix of sum and integral), in which case we will be dealing with a continous density (as oppose to discrete masses) for continuous random variable (or a mix of continuous and discrete parts).