هل من حلول لهذة المسائل ؟؟ Exercice 1 Let S be a set and let A = {A ⊂ S/A is countable or A c is countable}. Prove that A is a σ-algebra. Exercice 2 1. Suppose that Ai is a σ-algebra on S for every i ∈ I. Prove that ∩i∈IAi is a σ-algebra. 2. Give an example of two σ-algebras A and B on S = {1, 2, 3} such that A ∪ B is not a σ-algebra. Exercice 3 Let S1, S2 be two sets and f be a map from S1 to S2. 1. Prove that if B is a σ-algebra on S2, then A = f −1 (B) = {f −1 (B)/B ∈ B} is a σ-algebra on S1. 2. Prove that for any F2 ⊂ P(S2), we have
هل من حلول لهذة المسائل ؟؟
ردحذفExercice 1 Let S be a set and let
A = {A ⊂ S/A is countable or A
c
is countable}.
Prove that A is a σ-algebra.
Exercice 2
1. Suppose that Ai is a σ-algebra on S for every i ∈ I. Prove that ∩i∈IAi is a σ-algebra.
2. Give an example of two σ-algebras A and B on S = {1, 2, 3} such that A ∪ B is not a
σ-algebra.
Exercice 3 Let S1, S2 be two sets and f be a map from S1 to S2.
1. Prove that if B is a σ-algebra on S2, then
A = f
−1
(B) = {f
−1
(B)/B ∈ B}
is a σ-algebra on S1.
2. Prove that for any F2 ⊂ P(S2), we have