I am again tutoring the course MATH3029. The content is largely the same but the name has changed from “Probability Modelling and Applications” to “Probability Theory and Applications” to better reflect the material taught. There was a good question on the first assignment that leads to some interesting mathematics.
The assignment question is as follows. Let be a set and let be a -algebra on . Let be a function with the following properties
- For any finite sequence of pairwise disjoint sets in , we have .
- If is a sequence of sets in such that for all and , then, as tends to infinity, .
Students were then asked to show that the function is a probability measure on . This amounts to showing that is countably additive. That is if is a sequence of pairwise disjoint sets, then . One way to do this is define . Since the sets are pairwise disjoint, the sets satisfy the assumptions of the third property of . Thus we can conclude that as .
We also have that for every we have . Thus by applying the second property of twice we get
If we let tend to infinity, then we get the desired result.
A Follow Up Question
A natural follow up question is whether all three of the assumptions in the question are necessary. It is particularly interesting to ask if there is an example of a function that satisfies the first two properties but is not a probability measure. It turns out the answer is yes but coming up with an example involves some serious mathematics.
Let be the set of natural numbers and let be the power set of .
One way in which people talk about the size of a subset of natural numbers is to look at the proportion of elements in and take a limit. That is we could define
This function has some nice properties for instance if is the set of all even numbers than . More generally if is the set of all numbers divisible by , then . The function gets used a lot. When people say that almost all natural numbers satisfy a property, they normally mean that if is the subset of all numbers satisfying the property, then .
However the function is not a probability measure. The function is finitely additive. To see this, let be a finite collection of disjoint subsets of and let . Then for every natural number ,
Since the sets are disjoint, the union on the right is a disjoint union. Thus we have
Taking limits on both sides gives , as required. Furthermore, the function is not countably additive. For instance if we let for each . Then and . On the other hand for every and hence .
Thus it would appear that we have an example of a finitely additive measure that is not countably additive. However there is a big problem with the above definition of . Namely the limit of does not always exist. Consider the set , ie a number is in if and only if for some odd number . The idea with the set is that it looks a little bit like this:
There are chunks of numbers that alternate between being in and not being in and as we move further along, these chunks double in size. Let represent the sequence of numbers . We can see that increases while is in a chunk that belongs to and decreases when is in a chunk not in . More specifically if , then is increasing but if , then is decreasing.
At the turning points or we can calculate exactly what is equal to. Note that
Furthermore since there are no elements of between and we have
Thus we have
Hence the values fluctuate between approaching and . Thus the limit of does not exist and hence is not well-defined.
There is a work around using something called a Banach limit. Banach limits are a way of extending the notion of a limit from the space of convergent sequences to the space of bounded sequences. Banach limits aren’t uniquely defined and don’t have a formula describing them. Indeed to prove the existence of Banach limits one has to rely on non-constructive mathematics such as the Hanh-Banach extension theorem. So if we take for granted the existence of Banach limits we can define
where is now a Banach limit. This new definition of is defined on all subsets of and is an example of measure that is finitely additive but not countably additive. However the definition of is very non-constructive. Indeed there are models of ZF set theory where the Hanh-Banach theorem does not hold and we cannot prove the existence of Banach limits.
This begs the question of whether or not there exist constructible examples of measures that are finitely additive but not countably additive. A bit of Googling reveals that non-principal ultrafilters provide another way of defining non-countably additive measures. However the existence of a non-principal ultrafilter on is again equivalent to a weak form of the axiom of choice. Thus it seems that the existence of a non-countably additive measure may be inherently non-constructive. This discussion on Math Overflow goes into more detail.