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Countable additivity

Countable additivity is one of the properties that characterize probability.

Synonyms

Countable additivity is also called sigma-additivity ($\sigma $-additivity).

Statement

If $QTR{rm}{P}$ is a probability measure and [eq1] is a sequence of mutually exclusive events (i.e., [eq2] if $i\neq j$), then the following property, called countable additivity, holds:[eq3]

In other words, the probability of a union of disjoint events is equal to the sum of their probabilities.

It is easy to prove that countable additivity implies finite additivity, i.e.,[eq4]for any set of $N$ mutually exclusive events (just set $E_{n}=\emptyset $ for $n>N$ in the definition of countable additivity).

More details

More details about countable additivity - as well as about the other properties of probability - can be found in the lecture entitled Probability.

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