Slides Chapter 1. Measure Theory and Probability

1.1. SET SEQUENCES
When A n ⊇ A n+1 , ∀n ∈ IN, the sequence is said to be monotone
decreasing, and we represent it by {A n } ↓.
Example 1.1 Consider the sequences defined by:
(i) A n = (−n,n), ∀n ∈ IN. This sequence is monotone increasing,
since ∀n ∈ IN,
A n = (−n,n) ⊂ (−(n+1),n+1) = A n+1 .
(ii) B n = (−1/n,1+1/n), ∀n ∈ IN. This sequence is monotone
decreasing, since ∀n ∈ IN,
B n = (−1/n,1+1/n) ⊃ (−1/(n+1),1+1/(n+1)) = B n+1 .
Definition 1.3 (Limit of a set sequence)
(i) We call lower limit of {A n }, and we denote it limA n , to the
set of points of Ω that belong to all A n s except for a finite
number of them.
(ii) We call upper limit of {A n }, and we denote it limA n , to the
set of points of Ω that belong to infinite number of A n s. It
is also said that A n occurs infinitely often (i.o.), and it is
denoted also limA n = A n i.o.
Example 1.2 If ω ∈ A 2n , ∀n ∈ IN, then ω ∈ limA n but ω /∈
limA n since there is an infinite number of A n s to which ω does not
belong, {A 2n−1 } n∈IN .
Proposition 1.1 (Another characterization of limit of
a set sequence)
∞⋃ ∞⋂
(i) limA n =
k=1n=k
A n
ISABEL MOLINA 2