1.1. A review of probability theory 3
such that the whole space Ω has probability 1, and such that P is countably
additive.
Elements of the sample space Ω will be denoted ω. However, for reasons
that will be explained shortly, we will try to avoid actually referring to such
elements unless absolutely required to.
If we were studying just a single random process, e.g., rolling a single
die, then one could choose a very simple sample space; in this case, one
could choose the finite set {1,..., 6}, with the discrete σ-algebra
2{1,...,6}
:=
{A : A {1,..., 6}} and the uniform probability measure. But if one later
wanted to also study additional random processes (e.g., supposing one later
wanted to roll a second die, and then add the two resulting rolls), one would
have to change the sample space (e.g., to change it now to the product space
{1,..., 6} × {1,..., 6}). If one was particularly well organised, one could in
principle work out in advance all of the random variables one would ever
want or need, and then specify the sample space accordingly, before doing
any actual probability theory. In practice, though, it is far more convenient
to add new sources of randomness on the fly, if and when they are needed,
and extend the sample space as necessary. This point is often glossed over
in introductory probability texts, so let us spend a little time on it. We say
that one probability space , B , P )
extends1
another (Ω, B, P) if there is
a surjective map π : Ω Ω which is measurable (i.e.
π−1(E)
B for
every E B) and probability preserving (i.e. P
(π−1(E))
= P(E) for every
E B). By definition, every event E in the original probability space is
canonically identified with an event
π−1(E)
of the same probability in the
extension.
Example 1.1.1. As mentioned earlier, the sample space {1,..., 6}, that
models the roll of a single die, can be extended to the sample space {1,..., 6}
×{1,..., 6} that models the roll of the original die together with a new die,
with the projection map π : {1,..., 6} × {1,..., 6} {1,..., 6} being given
by π(x, y) := x.
Another example of an extension map is that of a permutation; for
instance, replacing the sample space {1,..., 6} by the isomorphic space
{a,...,f} by mapping a to 1, etc. This extension is not actually adding
any new sources of randomness, but is merely reorganising the existing ran-
domness present in the sample space.
1Strictly speaking, it is the pair ((Ω , B , P ), π) which is the extension of (Ω, B, P), not just
the space , B , P ), but let us abuse notation slightly here.
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