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Countable additivity

by , PhD

Countable additivity is one of the properties that characterize probability.

Table of Contents


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

The property

A well-defined probability measure $QTR{rm}{P}$ must have the property that [eq1]where [eq2] is a sequence of mutually exclusive events (i.e., [eq3] if $i\neq j$).

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

This property is called countable additivity.

Countable additivity implies finite additivity

It is easy to prove that countable additivity implies finite additivity, that is,[eq4]for any set of $N$ mutually exclusive events.


Note that [eq5] for any i. Therefore, we can set $E_{n}=\emptyset $ for $n>N$ in the definition of countable additivity.

How can it be proved?

People sometimes ask how countable additivity can be proved.

It cannot be proved. It is a defining property of probability. It is an axiom that $QTR{rm}{P}$ is required to satisfy in order to be called a probability.

Why is it required?

Why don't we impose the simpler and more easily understandable axiom of finite additivity?

Basically, it is for mathematical convenience.

With a slight abuse of notation, we would like the following continuity property of probability to hold[eq6]which allows us to interchange limits and probabilities.

This continuity is guaranteed only by the sigma-additivity axiom.

More details

The lecture on probability contains more details about countable additivity and the other two properties that define probability, namely:

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How to cite

Please cite as:

Taboga, Marco (2021). "Countable additivity", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix.

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