Theorem
measurability of limits
Let
X
1
,
X
2
,
…
X_1 , X_2 , \dots
be random variables defined on
(
Ω
,
ℬ
)
(\Omega, \mathcal{B})
. Then
∨
n
X
n
\vee_n X_n
and
∧
n
X
n
\wedge_n X_n
are random variables
lim inf
n
→
∞
X
n
\liminf_{n\to \infty}X_n
and
lim sup
n
→
∞
X
n
\limsup_{n\to \infty}X_n
are random variables
if for each
ω
\omega
the limit
lim
n
→
∞
X
n
(
ω
)
\lim_{n\to \infty}X_n (\omega)
exists then
lim
n
→
∞
X
n
\lim_{n\to \infty}X_n
is a random variable
the set on which
{
X
n
}
\{X_n\}
has a limit is measurable, i.e.
{
ω
:
lim
X
n
(
ω
)
exists
}
∈
ℬ
.
\{\omega: \lim X_n (\omega) \text{ exists} \} \in \mathcal{B} \text{.}