1 REVIEW OF MEASURE THEORY AND PROBABILITY THEORY 4
Then
P (A
UU
) = p
2
; P (A
UD
) = P (A
DU
) = pq; P (A
DD
) = q
2
.
Further, P (A
c
UU
) = 1 − p
2
; similarly, you can calculate the probability of each other
set in F
(2)
. Moreover, if A
UUU
= {ω : ω
1
= U, ω
2
= U, ω
3
= U}, you can calculate that
P (A
UUU
) = p
3
. And so on. Hence, in the limit you can conclude that the probability
of the sequence UUU... is zero. The same applies for example to the sequence UDUD...;
in fact this sequence is the intersection of the sequences U, UD, UDU, .... From this
example, we can conclude that every single sequence in Ω has probability zero.
In the previous example, we have shown that
P (every movement is up) = 0;
this implies that this event is sure not to happen. Similarly, since the above is true, we are
sure to get at least one down movement in the sequence, although we do not know exactly
when in the sequence. Because of this fact, and t he fact that the infinite sequence UUU...
is in the sample space (which means that still is a possible outcome), mathematicians have
come up with a somehow strange way of saying: we will get at least one down movement
almost surely.
Definition 6 Let (Ω, F, P) be a probability space. If A ⊂ F is such that
P (A) = 1,
we say that the event A occurs almost surely (a.s.).
Now, in order to introduce the next definition, consider the following, maybe a little
silly, example. Assume that you want to measure the length of a room, and assume you
express this measure in meters and centimeters. It turns out that the room is 4.30m.
long. Now assume that you want to change the reference system and express the length
of the room in terms of feet and inches. Then, the room is 14ft. long. But in the process
of switching from one reference system to the other, the room did not change: it did not
shrink; it did not expand. The same applies to events and probability measures. The idea
is given in the following.
Definition 7 (Absolutely continuous/equivalent probability measure) Give n two
probability measures P and P
∗
defined on the same σ-algebra F, then:
i) P is absolutely contin uous with respect to P
∗
, i.e. P << P
∗
, if P (A) = 0 whenever,
P
∗
(A) = 0∀A ∈ F.
ii) If P << P
∗
and also P
∗
<< P, then P ∼ P
∗
, i.e. P and P
∗
are equivalent measures.
Thus, for P ∼ P
∗
the following are equivalent:
• P (A) = 0 ⇔ P
∗
(A) = 0 (same null sets)
• P (A) = 1 ⇔ P
∗
(A) = 1 (same a.s. sets)
评论0
最新资源