Let A be a -field, and An A for all n N. Show that the following statements hold: (B4) A. (B5)

Let A be a σ-field, and An ∈ A for all n ∈ N. Show that the following statements hold:

(B4) ∅ ∈ A.

(B5)
Sn i=1 Ai ∈ A.
(B6)
Tn i=1 Ai ∈ A.
(B7)
T∞
i=1 Ai ∈ A.
(B8) lim supk→∞ Ak ∈ A.
(B9) lim infk→∞ Ak ∈ A.
Here, we define lim supk→∞ Ak ≡
T∞
n=1 S∞
k=n Ak and lim infk→∞ Ak ≡ S∞
n=1 T∞
k=n Ak.