Slides Chapter 1. Measure Theory and Probability

1.1. SET SEQUENCES
(ii) limA n =
∞⋂
∞⋃
k=1n=k
A n
Proposition 1.2 (Relationbetweenlower andupperlimits)
The lower and upper limits of a set sequence {A n } satisfy
limA n ⊆ limA n
Definition 1.4 (Convergence)
A set sequence {A n } converges if and only if
Then, we call limit of {A n } to
limA n = limA n .
lim
n→∞ A n = limA n = limA n .
Definition 1.5 (Inferior/Superior limit of a sequence of
real numbers)
Let {a n } n∈IN ∈ IR be a sequence. We define:
(i) liminf a n = sup inf a n;
n→∞ k n≥k
(ii) limsup
n→∞
a n = inf sup a n .
k n≥k
Proposition 1.3 (Convergence of monotoneset sequences)
Any monotone (increasing of decreasing) set sequence converges,
and it holds:
(i) If {A n } ↑, then lim
n→∞
A n =
(ii) If {A n } ↓, then lim
n→∞
A n =
∞⋃
A n .
n=1
∞⋂
A n .
n=1
ISABEL MOLINA 3