Theorem
the eight properties of probability measures
complements
:
ℙ
(
A
∁
)
=
1
−
ℙ
(
A
)
\mathbb{P}(A^{\complement}) = 1 - \mathbb{P}(A)
empty
:
ℙ
(
∅
)
=
0
\mathbb{P}(\emptyset) = 0
inclusion-exclusion
:
ℙ
(
A
∪
B
)
=
ℙ
(
A
)
+
ℙ
(
B
)
−
ℙ
(
A
∩
B
)
\mathbb{P}(A \cup B) = \mathbb{P}(A) + \mathbb{P}(B) - \mathbb{P}(A \cap B)
monotonicity
:
A
⊂
B
⟹
ℙ
(
A
)
≤
ℙ
(
B
)
A \subset B \implies \mathbb{P}(A) \leq \mathbb{P}(B)
subadditivity
:
ℙ
(
∪
n
=
1
∞
A
n
)
≤
Σ
n
=
1
∞
ℙ
(
A
n
)
\mathbb{P}(\cup_{n=1}^{\infty}A_n) \leq \Sigma_{n=1}^{\infty}\mathbb{P}(A_n)
continuity
: if
A
n
↑
A
,
A
n
∈
ℬ
A_n \uparrow A, A_n \in \mathcal{B}
then
ℙ
(
A
n
)
↑
ℙ
(
A
)
\mathbb{P}(A_n) \uparrow \mathbb{P}(A)
; if
A
n
↓
A
,
A
n
∈
ℬ
A_n \downarrow A, A_n \in \mathcal{B}
then
ℙ
(
A
n
)
↓
ℙ
(
A
)
\mathbb{P}(A_n) \downarrow \mathbb{P}(A)
Fatou
: if
A
n
∈
ℬ
A_n \in \mathcal{B}
for
n
≥
1
n \geq 1
then
ℙ
(
lim inf
A
n
)
≤
lim inf
ℙ
(
A
n
)
≤
lim sup
ℙ
(
A
n
)
≤
ℙ
(
lim sup
A
n
)
\mathbb{P}(\liminf A_n) \leq \liminf \mathbb{P}(A_n) \leq \limsup \mathbb{P}(A_n) \leq \mathbb{P}(\limsup A_n)
more continuity
: if
lim
A
n
=
A
\lim A_n = A
then
lim
ℙ
(
A
n
)
=
ℙ
(
A
)
\lim \mathbb{P}(A_n) = \mathbb{P}(A)