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Tableau 5.
Towards an axiomatic theory of probability: The abstract
probability space.
Having explored chance in commonplace settings to build intuition and
having gone through and reviewed the language of set theory
with a view to giving us an alphabet, we have now come to the point
where we can build a formal mathematical foundation for the subject.
This is very much in the way that in Euclidean geometry,
people built upon the intuition for geometrical concepts from antiquity.
And about 2 million year ago, a scribe in Alexandria
put forth a formal axiomatic foundation for the subject.
And this of course revolutionized the world.
This was Euclidean geometry.
The foundations for probability should follow a similar approach.
But, remarkably for a subject of such antiquity, the formal foundations
were only laid less than a century ago, in the early part of the 20th century.
So, let us begin with our understanding of ideas that we have gleaned from chance
in ordinary settings, to see what should go into the foundation of the subject.
So, to begin,
each of the experiments we looked at had three features.
There was an idea that when you perform a chance experiment, there is an outcome,
and the outcomes together comprise a family of conceivable outcomes.
Second, there was the idea of an event of interest.
And we identified events with aggregates, collections of outcomes.
And finally, there was a process of ascribing a chance to an event,
a collection of outcomes.
In the formal theory, each of these elements has a formal counterpart.
The collection of outcomes is called the sample space.
The family of events are called the family of measurable sets.
And finally, the concept of chance is codified
in the provision of a probability measure.
In the usual nomenclature, we give some notational names to these objects.
The sample space is generally represented by the uppercase Greek letter omega.
The family of events of interest to us is usually represented by a script f.
And the probability measure is represented inevitably by the letter P in bold font.
These, if you wish, you can think of as the Probabilist's Trinity.