Slides Chapter 1. Measure Theory and Probability
Slides Chapter 1. Measure Theory and Probability
Slides Chapter 1. Measure Theory and Probability
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<strong>1.</strong><strong>1.</strong> SET SEQUENCES<br />
When A n ⊇ A n+1 , ∀n ∈ IN, the sequence is said to be monotone<br />
decreasing, <strong>and</strong> we represent it by {A n } ↓.<br />
Example <strong>1.</strong>1 Consider the sequences defined by:<br />
(i) A n = (−n,n), ∀n ∈ IN. This sequence is monotone increasing,<br />
since ∀n ∈ IN,<br />
A n = (−n,n) ⊂ (−(n+1),n+1) = A n+1 .<br />
(ii) B n = (−1/n,1+1/n), ∀n ∈ IN. This sequence is monotone<br />
decreasing, since ∀n ∈ IN,<br />
B n = (−1/n,1+1/n) ⊃ (−1/(n+1),1+1/(n+1)) = B n+1 .<br />
Definition <strong>1.</strong>3 (Limit of a set sequence)<br />
(i) We call lower limit of {A n }, <strong>and</strong> we denote it limA n , to the<br />
set of points of Ω that belong to all A n s except for a finite<br />
number of them.<br />
(ii) We call upper limit of {A n }, <strong>and</strong> we denote it limA n , to the<br />
set of points of Ω that belong to infinite number of A n s. It<br />
is also said that A n occurs infinitely often (i.o.), <strong>and</strong> it is<br />
denoted also limA n = A n i.o.<br />
Example <strong>1.</strong>2 If ω ∈ A 2n , ∀n ∈ IN, then ω ∈ limA n but ω /∈<br />
limA n since there is an infinite number of A n s to which ω does not<br />
belong, {A 2n−1 } n∈IN .<br />
Proposition <strong>1.</strong>1 (Another characterization of limit of<br />
a set sequence)<br />
∞⋃ ∞⋂<br />
(i) limA n =<br />
k=1n=k<br />
A n<br />
ISABEL MOLINA 2