Probability of Continuous Random Variable Lying in Singleton Set is Zero/Corollary

Theorem
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be a continuous real variable on $\struct {\Omega, \Sigma, \Pr}$. Let $C$ be a countable subset of $\R$.

Then:


 * $\map \Pr {X \in C} = 0$

Proof
Since $C$ is countable, there exists a sequence $\sequence {x_n}_{n \mathop \in \N}$ of distinct real numbers such that:


 * $C = \set {x_n : n \mathop \in \N}$

That is:


 * $\ds C = \bigcup_{n \mathop = 1}^\infty \set {x_n}$

where $\set {\set {x_1}, \set {x_2}, \ldots}$ is pairwise disjoint.

We then have: