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 (Ω , 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 (Ω , B , P ), but let us abuse notation slightly here.