Last semester I tutored the course Probability Modelling with Applications. In this course the main objects of study are probability spaces. A ** probability space** is a triple where:

- is a set.
- is a -algebra on . That is is a collection of subsets of such that and is closed under set complements and countable unions. The element of are called events and they are precisely the subsets of that we can assign probabilities to. We will denote the power set of by and hence .
- is a probability measure. That is it is a function such that and for all countable collections of mutually disjoint subsets we have that .

It’s common for students to find probability spaces, and in particular -algebras, confusing. Unfortunately Vitalli showed that -algebras can’t be avoided if we want to study probability spaces such as or an infinite number of coin tosses. One of the main reasons why -algebras can be so confusing is that it can be very hard to give concrete descriptions of all the elements of a -algebra.

We often have a collection of subsets of that we are interested in but this collection fails to be a -algebra. For example, we might have and is the collection of open subsets. In this situation we take our -algebra to be which is the smallest -algebra containing . That is

where the above intersection is taken over all -algebras that contain . In this setting we will say that generates . When we have such a collection of generators, we might have an idea for what probability we would like to assign to sets in . That is we have a function and we want to extend this function to create a probability measure . A famous theorem due to Caratheodory shows that we can do this in many cases.

An interesting question is whether the extension is unique. That is does there exists a probability measure on such that but ? The following theorem gives a criterion that guarantees no such exists.

**Theorem: **Let be a set and let be a collection of subsets of that is closed under finite intersections. Then if are two probability measures such that , then .

The above theorem is very useful for two reasons. Firstly it can be combined with Caratheodory’s extension theorem to uniquely define probability measures on a -algebra by specifying the values on a collection of simple subsets . Secondly if we ever want to show that two probability measures are equal, the above theorem tells us we can reduce the problem to checking equality on the simpler subsets in .

The condition that must be closed under finite intersections is somewhat intuitive. Suppose we had but . We will however have and thus we might be able to find two probability measure such that and but . The following counterexample shows that this intuition is indeed well-founded.

When looking for examples and counterexamples, it’s good to try to keep things as simple as possible. With that in mind we will try to find a counterexample when is finite set with as few elements as possible and is equal to the powerset of . In this setting, a probability measure can be defined by specifying the values for each .

We will now try to find a counterexample when is as small as possible. Unfortunately we won’t be able find a counterexample when only contains one or two elements. This is because we want to find such that is not equal to or .

Thus we will start out search with a three element set . Up to relabelling the elements of , the only interesting choice we have for is . This has a chance of working since is not closed under intersection. However any probability measure on must satisfy the equations

- ,
- ,
- .

Thus , and . Thus is determined by its values on and .

However, a four element set is sufficient for our counter example! We can let . Then and we can define by

- , , and .
- , , and .

Clearly however and . Thus we have our counterexample! In general for any we can define the probability measure . The measure is not equal to but agrees with on . In general, if we have two probability measures that agree on but not on then we can produce uncountably many such measures by taking convex combinations as done above.

Don’t have much to say about this. You did a terrific job man! Probably what I can do is add a reference? So if this topic is of your interests, either talk to Micheal (preferred one) or check “Probability and Measure by Billingsley” or “Real Analysis and Probability by Dudley” for more details.

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Good point! For the course we used the textbook “Stochastics : Introduction to Probability and Statistics” by Hans Otto Georgii.

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