Slides Chapter 1. Measure Theory and Probability

1.4. PROBABILITY MEASURES
defined on an algebra A, then for all A 1 ,...,A n ∈ A,
( n
)
⋃ n∑
µ A i ≤ µ(A i ).
i=1
5. Prove that for any measure µ defined on an algebra A, then
for all A 1 ,...,A n ∈ A such that ⋃ ∞
n=1 A n ∈ A,
( ∞
)
⋃ ∞∑
µ A n ≤ µ(A n ).
n=1
1.4 Probability measures
i=1
n=1
Definition 1.19 (Random experiment)
A random experiment is a process for which:
• the set of possible results is known;
• its result cannot be predicted without error;
• if we repeat it in identical conditions, the result can be different.
Definition 1.20 (Elementary event, sample space, event)
The possible results of the random experiment that are indivisible
are called elementary events. The set of elementary events is
known as sample space, and it will be denoted Ω. An event A is a
subset of Ω, such that once the random experiment is carried out,
we can say that A “has occurred” if the result of the experiment
is contained in A.
Example 1.8 Examples of random experiments are:
(a) Tossing a coin. The sample space is Ω = {“head”,“tail”}.
Events are: ∅, {“head”}, {“tail”}, Ω.
ISABEL MOLINA 12