# Probability Measure on Single-Subset Event Space

## Theorem

Let $\EE$ be an experiment whose sample space is $\Omega$.

Let $\O \subsetneqq A \subsetneqq \Omega$.

Let $\Sigma := \set {\O, A, \Omega \setminus A, \Omega}$ be the event space of $\EE$.

Let $\Pr: \Sigma \to \R$ be a probability measure on $\struct {\Omega, \Sigma}$.

Then $\Pr$ has the form:

\(\text {(Pr 1)}: \quad\) | \(\ds \map \Pr \O\) | \(=\) | \(\ds 0\) | |||||||||||

\(\text {(Pr 2)}: \quad\) | \(\ds \map \Pr A\) | \(=\) | \(\ds p\) | |||||||||||

\(\text {(Pr 3)}: \quad\) | \(\ds \map \Pr {\Omega \setminus A}\) | \(=\) | \(\ds 1 - p\) | |||||||||||

\(\text {(Pr 4)}: \quad\) | \(\ds \map \Pr \Omega\) | \(=\) | \(\ds 1\) |

for some $p \in \R$ satisfying $0 \le p \le 1$.

## Proof

From Event Space from Single Subset of Sample Space, we have that $\Sigma$ is an event space.

Recall the Kolmogorov axioms:

\((1)\) | $:$ | \(\ds \forall A \in \Sigma:\) | \(\ds 0 \) | \(\ds \le \) | \(\ds \map \Pr A \le 1 \) | The probability of an event occurring is a real number between $0$ and $1$ | |||

\((2)\) | $:$ | \(\ds \map \Pr \Omega \) | \(\ds = \) | \(\ds 1 \) | The probability of some elementary event occurring in the sample space is $1$ | ||||

\((3)\) | $:$ | \(\ds \map \Pr {\bigcup_{i \mathop \ge 1} A_i} \) | \(\ds = \) | \(\ds \sum_{i \mathop \ge 1} \map \Pr {A_i} \) | where $\set {A_1, A_2, \ldots}$ is a countable (possibly countably infinite) set of pairwise disjoint events | ||||

That is, the probability of any one of countably many pairwise disjoint events occurring | |||||||||

is the sum of the probabilities of the occurrence of each of the individual events |

First we determine that $\Pr$ as defined is actually a probability measure.

Axiom $(1)$ and axiom $(2)$ are fulfilled trivially by definition.

Then we note that, apart from $\O$, $\set {A, \Omega \setminus A}$ are the only pairwise disjoint events whose union is $\Omega$.

Hence by definition of $\Pr$ we see that axiom $(3)$ degenerates to:

\(\ds \map \Pr {A \cup \paren {\Omega \setminus A} }\) | \(=\) | \(\ds \map \Pr \Omega\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 1\) | $\text {Pr 4}$ | |||||||||||

\(\ds \) | \(=\) | \(\ds p + \paren {1 - p}\) | Definition of $p$ | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \Pr A + \map \Pr {\Omega \setminus A}\) | $\text {Pr 2}$ and $\text {Pr 3}$ |

and thus axiom $(3)$ is fulfilled.

Thus $\Pr$ is indeed a probability measure.

$\Box$

Now it is established that a probability measure on $\Sigma$ is of the form of $\Pr$ as defined.

First we note that from Probability of Empty Event is Zero:

- $\map \Pr \O = 0$

Hence $\text {Pr 1}$ is satisfied.

From Axiom $(2)$ we have that $\map \Pr \Omega = 1$

Hence $\text {Pr 4}$ is satisfied.

By definition of probability measure, $\map \Pr A$ is between $0$ and $1$ inclusive.

That is:

- $\map \Pr A = p$

for some $p \in \R$ such that $0 \le p \le 1$.

This is exactly $\text {Pr 2}$, which is hence seen to be satisfied.

Then we note that from Probability of Event not Occurring:

- $\map \Pr {\Omega \setminus A} = 1 - \map \Pr A$

That is:

- $\map \Pr {\Omega \setminus A} = 1 - p$

Hence $\text {Pr 3}$ is satisfied.

Thus it has been proved that a probability measure on such an an event space $\Sigma$ has to be of the form $\Pr$ as defined.

$\blacksquare$

## Proof

- 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $1$: Events and probabilities: $1.3$: Probabilities: Example $10$