Finite Additive Probability
Countable Additivity and Finite Additivity
In our textbook, the axioms of probability, Definition 2.6, are given below. The Axiom 3 is countable additivity (CA) which we call countable additive probability (CAP).
In class, we rised the discussion on what if we only allow finite additivity? Which means we modify the Axiom 3 to
- Axiom 3’: If \(A_1, A_2, \ldots, A_k\) forms a sequence of pairwise mutually exclusive events in \(S\) (that is, \(A_i\cap A_j = \emptyset\) if \(i\neq j\)), then \[P\left(A_1 \cup A_2 \cup A_3 \cup \cdots \cup A_k\right)=\sum_{i=1}^{k} P\left(A_i\right).\]
Therefore, we will have Finite Additivity Probability (FAP) which Bruno de Finetti is the most famous supporter on FAP.
We can think about what the relation and difference between them.
Axioms cannot be right or wrong for a theory. They are just assumptions, like parallel postulate in geometry.
Countable additivity contains finite additivity as a special case (thinking the countable sequence of \(A_1, A_2, \ldots,\) only has finite distinct events). But in FAP, it can express concepts that cannot be expressed by CAP.
Example Consider the natural numbers \(S=\{1,2,3, \ldots,\}\) as a sample space. There is no uniform distribution on \(S\) in CAP, because no matter how to assign the probability to each point, say \(P(\{i\})=a\) for \(i \in S\). It will lead a contradiction. In particularly, if \(a=0\), then apply CA, \(P(S)=0\) which contradicts the normalization axiom, \(P(S)=1\). If \(a>0\), then \(P(S)=\infty\) which is still not compatible with normalization axiom.
But, there are finite additive probabilities on \(S\). For example, we let \(P(\{i\})=0\) for \(i \in S\) which will NOT contradict the normalization axiom. Also, we can assign probability 0.5 to the even numbers and odd numbers. Again this does not conflict with each integer having probability 0 as long as you do not insist on countable additivity.
Why CAP?
Above example shows some good features of FAP, but why our textbook takes CAP. Again, there is no right or wrong on a system of axioms, but some systems are more useful. Countable additivity is undoubtedly useful which is one pillar of mathematical analysis. Denying CA means we only use rational numbers rather than real numbers. For probability theory, there is an established results (Schervish, Seidenfeld, and Kadane 1984) that the law of total probability \[P(A)=\sum_j P\left(A \mid B_j\right) P\left(B_j\right)\] will not hold in general. If anyone are interested in deeper results, the concept non-conglomerability is a good starting point.