Probability of Continuous Random Variable Lying in Singleton Set is Zero

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Theorem

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

Let $X$ be a continuous real variable on $\struct {\Omega, \Sigma, \Pr}$.


Then, for each $x \in \R$, we have:

$\map \Pr {X \le x} = \map \Pr {X < x}$

In particular:

$\map \Pr {X = x} = 0$


Corollary

Let $C$ be a countable subset of $\R$.


Then:

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


Proof

Let $F_X$ be the cumulative distribution function of $X$ so that:

$\map {F_X} x = \map \Pr {X \le x}$

for each $x \in \R$.

Let $P_X$ be the probability distribution of $X$.

Since $X$ is a continuous real variable, we have:

$F_X$ is continuous.

From Sequential Continuity is Equivalent to Continuity in the Reals, we have:

for each real sequence $\sequence {x_n}_{n \mathop \in \N}$ converging to $x$, we have $\map {F_X} {x_n}\to \map {F_X} x$.

For each $n \in \N$, let:

$\ds x_n = x - \frac 1 n$

We have that $x_n \to x$, so:

$\ds \map {F_X} x = \lim_{n \mathop \to \infty} \map {F_X} {x - \frac 1 n}$

We also have that $\sequence {x_n}_{n \mathop \in \N}$ is a increasing sequence.

So, we have:

$\ds \hointl {-\infty} {x - \frac 1 n} \subseteq \hointl {-\infty} {x - \frac 1 {n + 1} }$

for each $n \in \N$.

So:

$\ds \sequence {\hointl {-\infty} {x - \frac 1 n} }_{n \mathop \in \N}$ is a increasing sequence of sets.

We can see that:

$\ds \bigcup_{n \mathop = 1}^\infty \hointl {-\infty} {x - \frac 1 n} = \openint {-\infty} x$

Lemma

Let $x$ be a real number.

Then:

$\ds \bigcup_{n \mathop = 1}^\infty \hointl {-\infty} {x - \frac 1 n} = \openint {-\infty} x$

$\Box$


So, we have:

\(\ds \map \Pr {X < x}\) \(=\) \(\ds \map {P_X} {\openint {-\infty} x}\) Definition of Probability Distribution
\(\ds \) \(=\) \(\ds \lim_{n \mathop \to \infty} \map {P_X} {\hointl {-\infty} {x - \frac 1 n} }\) Measure of Limit of Increasing Sequence of Measurable Sets
\(\ds \) \(=\) \(\ds \lim_{n \mathop \to \infty} \map \Pr {X \le x - \frac 1 n}\) Definition of Probability Distribution
\(\ds \) \(=\) \(\ds \lim_{n \mathop \to \infty} \map {F_X} {x - \frac 1 n}\) Definition of Cumulative Distribution Function
\(\ds \) \(=\) \(\ds \map {F_X} x\)
\(\ds \) \(=\) \(\ds \map \Pr {X \le x}\) Definition of Cumulative Distribution Function

We then have:

\(\ds \map \Pr {X \le x}\) \(=\) \(\ds \map \Pr {\set {X = x} \cup \set {X < x} }\)
\(\ds \) \(=\) \(\ds \map \Pr {X = x} + \map \Pr {X < x}\) Probability of Union of Disjoint Events is Sum of Individual Probabilities

giving:

$\map \Pr {X = x} = 0$

$\blacksquare$


Also see

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