Probability Measure

Definition

Probability Measure

A probability measure over a sigma algebra is a measure which assigns every possible event a probability.

A probability must satisfy all three Kolmogorov axioms.

Kolmogorov Axioms

Definition

Kolmogorov Axioms

Let be a probability space. The following three axioms are satisfied by any probability space:

Definition

Non-negativity ( Kolmogorov Axioms)

A probability measure satisfies non-negativity, i.e.,

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Definition

Normalisation ( Kolmogorov Axioms)

A probability measure satisfies normalisation, i.e.,

where is the sample space.

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Definition

Countable Sigma-additivity ( Kolmogorov Axioms)

A probability measure satisfies countable -additivity, i.e.,

where disjoint refers to disjoint events.

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