4 1. Preparatory material

In order to have the freedom to perform extensions every time we need

to introduce a new source of randomness, we will try to adhere to the fol-

lowing important

dogma2:

probability theory is only “allowed” to

study concepts and perform operations which are preserved with

respect to extension of the underlying sample space. As long as one

is adhering strictly to this dogma, one can insert as many new sources of

randomness (or reorganise existing sources of randomness) as one pleases;

but if one deviates from this dogma and uses specific properties of a single

sample space, then one has left the category of probability theory and must

now take care when doing any subsequent operation that could alter that

sample space. This dogma is an important aspect of the probabilistic way

of thinking, much as the insistence on studying concepts and performing

operations that are invariant with respect to coordinate changes or other

symmetries is an important aspect of the modern geometric way of think-

ing. With this probabilistic viewpoint, we shall soon see the sample space

essentially disappear from view altogether, after a few foundational issues

are dispensed with.

Let us now give some simple examples of what is and what is not a

probabilistic concept or operation. The probability P(E) of an event is a

probabilistic concept; it is preserved under extensions. Similarly, Boolean

operations on events such as union, intersection, and complement are also

preserved under extensions and are thus also probabilistic operations. The

emptiness or non-emptiness of an event E is also probabilistic, as is the

equality or

non-equality3

of two events E, F . On the other hand, the car-

dinality of an event is not a probabilistic concept; for instance, the event

that the roll of a given die gives 4 has cardinality one in the sample space

{1,..., 6}, but has cardinality six in the sample space {1,..., 6}×{1,..., 6}

when the values of an additional die are used to extend the sample space.

Thus, in the probabilistic way of thinking, one should avoid thinking about

events as having cardinality, except to the extent that they are either empty

or non-empty. For a related reason, the notion of the underlying probability

space being complete (i.e. every subset of a null set is again a null set) is not

preserved by extensions, and is thus technically not a probabilistic notion.

As such, we will downplay the role of completeness in our probability spaces.

Indeed, once one is no longer working at the foundational level, it is best

to try to suppress the fact that events are being modeled as sets altogether.

2This

is analogous to how differential geometry is only “allowed” to study concepts and

perform operations that are preserved with respect to coordinate change, or how graph theory

is only “allowed” to study concepts and perform operations that are preserved with respect to

relabeling of the vertices, etc..

3Note how it was important here that we demanded the map π to be surjective in the

definition of an extension.