Slides Chapter 1. Measure Theory and Probability

1.4. PROBABILITY MEASURES
For this, we will first consider the collection of intervals
C = {(a,b],(−∞,b],(a,+∞) : a < b}. (1.1)
We start by assigning values of P to intervals from C, by ensuring
that P is σ-additive on C. Then, we consider the algebra F obtained
by doing finite unions of disjoint intervals from C and we
extend P to F. The extended function will be a probability measureon(IR,F).
Finally,thereisauniqueextensionofaprobability
measure from F to σ(F) = B, see the following propositions.
Proposition 1.7 Consider the collection of all finite unions of
disjoint intervals from C in (1.1),
{ n
}
⋃
F = A i : A i ∈ C, A i disjoint .
Then F is an algebra.
i=1
Next we extend P from C to F as follows.
Proposition 1.8 (Extensionof theprobabilityfunction)
(a) For all A ∈ F, since A = ⋃ n
i=1 (a i,b i ], with a i ,b i ∈ IR ∪
{−∞,+∞}, let us define
n∑
P 1 (A) = P(a i ,b i ].
i=1
Then, P 1 is a probability measure over (IR,F).
(b) For all A ∈ C, it holds that P(A) = P 1 (A).
Observe that B = σ(F). Finally, can we extend P from F to
B = σ(F)? If the answer is positive, is the extension unique? The
next theorem gives the answer to these two questions.
ISABEL MOLINA 17