Probability of Continuous Random Variable Lying in Singleton Set is Zero/Corollary
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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:
\(\ds \map \Pr {X \in C}\) | \(=\) | \(\ds \map {P_X} C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map {P_X} {\bigcup_{n \mathop = 1}^\infty \set {x_n} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \map {P_X} {\set {x_n} }\) | using the countable additivity of $P_X$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \map \Pr {X \in \set {x_n} }\) | Definition of Probability Distribution | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \map \Pr {X = x_n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0\) | Probability of Continuous Random Variable Lying in Singleton Set is Zero |
$\blacksquare$