Equivalence of Definitions of Probability Measure

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Theorem

Let $\EE$ be an experiment.

The following definitions of the concept of Probability Measure are equivalent:

Definition 1

Let $\EE$ be defined as a measure space $\struct {\Omega, \Sigma, \Pr}$.

Then $\Pr$ is a measure on $\EE$ such that $\map \Pr \Omega = 1$.

Definition 2

Let $\Omega$ be the sample space on $\EE$.

Let $\Sigma$ be the event space of $\EE$.


A probability measure on $\EE$ is a mapping $\Pr: \Sigma \to \R$ which fulfils 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


Definition 3

Let $\Omega$ be the sample space on $\EE$.

Let $\Sigma$ be the event space of $\EE$.


A probability measure on $\EE$ is a mapping $\Pr: \Sigma \to \R$ which fulfils the following axioms:


\((\text I)\)   $:$     \(\ds \forall A \in \Sigma:\)    \(\ds \map \Pr A \)   \(\ds \ge \)   \(\ds 0 \)             
\((\text {II})\)   $:$      \(\ds \map \Pr \Omega \)   \(\ds = \)   \(\ds 1 \)             
\((\text {III})\)   $:$     \(\ds \forall A \in \Sigma:\)    \(\ds \map \Pr A \)   \(\ds = \)   \(\ds \sum_{\bigcup \set e \mathop = A} \map \Pr {\set e} \)             where $e$ denotes the elementary events of $\EE$

Definition 4

Let $\Omega$ be the sample space on $\EE$.

Let $\Sigma$ be the event space of $\EE$.


A probability measure on $\EE$ is an additive function $\Pr: \Sigma \to \R$ which fulfils the following axioms:


\((1)\)   $:$     \(\ds \forall A, B \in \Sigma: A \cap B = \O:\)    \(\ds \map \Pr {A \cup B} \)   \(\ds = \)   \(\ds \map \Pr A + \map \Pr B \)             
\((2)\)   $:$      \(\ds \map \Pr \Omega \)   \(\ds = \)   \(\ds 1 \)             


Proof