Definition:Probability Measure

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[edit] Context

Probability Theory.


[edit] Definition

Let \mathcal E be an experiment.

Let \Omega be the sample space on \mathcal E, and let \Sigma be the event space of \mathcal E.


A probability measure on \mathcal E is a mapping \Pr: \Sigma \to \R which fulfils the Kolmogorov axioms:


[edit] First Axiom

\forall A \in \Sigma: 0 \le \Pr \left({A}\right)


[edit] Second Axiom

\Pr \left({\Omega}\right) = 1


[edit] Third Axiom

Let A_1, A_2, \ldots be a countable (possibly countably infinite) sequence of pairwise disjoint events.

Then:

\Pr \left({\bigcup_{i \ge 1} A_i}\right) = \sum_{i \ge 1} \Pr \left({A_i}\right)


As an elementary an easily-digested consequence of this, we have:

\forall A, B \in \Sigma: A \cap B = \varnothing \implies \Pr \left({A \cup B}\right) = \Pr \left({A}\right) + \Pr \left({B}\right).


[edit] Notes

From the definition of event space, we already have that \varnothing \in \Sigma and \Omega \in \Sigma.


If \mathcal E is defined as being a measure space \left({\Omega, \Sigma, \Pr}\right), then \Pr is a measure on \mathcal E such that \Pr \left({\Omega}\right) = 1.


Also see Elementary Properties of Probability Measure for further immediate consequences of this definition.

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