Slides Chapter 1. Measure Theory and Probability

1.1. SET SEQUENCES
Example 1.3 Obtain the limits of the following set sequences:
(i) {A n }, where A n = (−n,n), ∀n ∈ IN.
(ii) {B n }, where B n = (−1/n,1+1/n), ∀n ∈ IN.
(i) By the previous proposition, since {A n } ↑, then
∞⋃ ∞⋃
lim A n = A n = (−n,n) = IR.
n→∞
n=1
n=1
(ii) Again, using the previous proposition, since {A n } ↓, then
∞⋂ ∞⋂
(
lim B n = B n = − 1
n→∞ n ,1+ 1 )
= [0,1].
n
Problems
n=1
1. Prove Proposition 1.1.
n=1
2. Define sets of real numbers as follows. Let A n = (−1/n,1] if
n is odd, and A n = (−1,1/n] if n is even. Find limA n and
limA n .
3. Prove Proposition 1.2.
4. Prove Proposition 1.3.
5. Let Ω = IR 2 and A n the interior of the circle with center at
the point ((−1) n /n,0) and radius 1. Find limA n and limA n .
6. Prove that (limA n ) c = limA c n and (limA n ) c = limA c n.
7. Using the De Morgan laws and Proposition 1.3, prove that if
{A n } ↑ A, then {A c n} ↓ A c while if {A n } ↓ A, then {A c n} ↑
A c .
ISABEL MOLINA 4