Countable additivity is one of the properties that characterize probability.
Countable additivity is also called sigma-additivity
(
-additivity).
A well-defined probability measure
must
have the property that
![[eq1]](/images/countable-additivity__3.png)
where
![[eq2]](/images/countable-additivity__4.png){kind=small}
is a sequence of mutually exclusive events (i.e.,
![[eq3]](/images/countable-additivity__5.png){kind=small}
if
).
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.
It is easy to prove that countable additivity implies finite additivity, that
is,![[eq4]](/images/countable-additivity__7.png)
for
any set of
mutually exclusive events.
Note that
![[eq5]](/images/countable-additivity__9.png){kind=small}
for any
.
Therefore, we can set
for
in the definition of countable additivity.
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
is required to satisfy in order to be called a probability.
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]](/images/countable-additivity__14.png)
which
allows us to interchange limits and probabilities.
This continuity is guaranteed only by the sigma-additivity axiom.
The lecture on probability contains more details about countable additivity and the other two properties that define probability, namely:
the range property,
the sure thing property.
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Please cite as:
Taboga, Marco (2021). "Countable additivity", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/glossary/countable-additivity.
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