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:

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

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:

$\ds \map \Pr {X \in C}$

$=$

$\ds \map {P_X} C$

$\ds $

$=$

$\ds \map {P_X} {\bigcup_{n \mathop = 1}^\infty \set {x_n} }$

$\ds $

$=$

$\ds \sum_{n \mathop = 1}^\infty \map {P_X} {\set {x_n} }$

$\ds $

$=$

$\ds \sum_{n \mathop = 1}^\infty \map \Pr {X \in \set {x_n} }$

$\ds $

$=$

$\ds \sum_{n \mathop = 1}^\infty \map \Pr {X = x_n}$

$\ds $

$=$

$\ds 0$

$\blacksquare$