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.
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