Show that the Borel algebra B is also generated by the collection of all half-openintervals (a, b] = {x ∈ R | a < x ≤ b}. Also show that B is generated by thecollection of all half-rays {x ∈ R | x > a}, a ∈ R.7. (a) Let E ⊂ R. Show that F = {∅, E, R \ E, R} is the σ-algebra of subsets of Rgenerated by {E}.(b) If S and T are collections of subsets of R, thenσ(S ∪ T ) = σ(S) ∪ σ(T ).Is the statement true? Why?Note: σ(M) is a σ-algebra generated by the collection M of some subsets.8. Consider F = {E ⊂ R | E is countable or R \ E is countable }.(a) Show that F is a σ-algebra and F is a proper sub-σ-algebra of the σ-algebra BRwhich is the Borel algebra of subsets of R.(b) Show that F is the σ-algebra generated by {{x} | x ∈ R}.9. Suppose X is a non-empty set and A = {A1, A2, . . . , Ak}, where the sets A1, . . . , Akare nonempty and form a partition of X, i.e. they are pairwise disjoint and ⋃ki=1 Ai =X. Show that σ(A) = {⋃j∈J Aj | J ⊆ {1, 2, . . . , k}}, where σ(A) is a σ-algebragenerated by A.10. If B(R) is the Borel σ-algebra of R and σ(I) is a σ-algebra generated by I, show thatB(R) = σ(I) if I = {[a, b) | a, b ∈ R, a ≤ b}.11. A nonempty collection M of subsets of a set X is called a monotone class if, for eachmonotone increasing sequence (En) in M and each monotone decreasing sequence(An) in M, the sets ⋃∞n=1 En and ⋂∞n=1 An belong to M.Show that a σ-algebra F of subsets of X is a monotone class.

This is a complex set of questions, so let's break it down step by step:

1. The Borel algebra B is generated by the collection of all open intervals (a, b). To show that it is also generated by all half-open intervals (a, b] we need to show that every open interval can be expressed as a countable union of half-open intervals. For any open interval (a, b), we can express it as the union of the half-open intervals (a, b - 1/n] for all n in the set of natural numbers. This shows that the Borel algebra can be generated by the half-open intervals.

2. To show that B is generated by the collection of all half-rays {x ∈ R | x > a}, a ∈ R, we need to show that every open interval can be expressed as a countable union or intersection of these half-rays. For any open interval (a, b), we can express it as the intersection of the half-rays {x ∈ R | x > a} and the complement of the half-ray {x ∈ R | x > b}. This shows that the Borel algebra can be generated by these half-rays.

3. (a) The σ-algebra generated by a set {E} is the collection of all subsets that can be formed from E using the operations of union, intersection, and complement. In this case, the only subsets that can be formed from E are the empty set, E itself, the complement of E, and the whole set R. Therefore, F = {∅, E, R \ E, R} is the σ-algebra generated by {E}.

   (b) The statement is not true. The σ-algebra generated by the union of two collections S and T is not necessarily the union of the σ-algebras generated by S and T. This is because the σ-algebra generated by a collection includes not just the sets in the collection, but also all possible unions, intersections, and complements of these sets. Therefore, σ(S ∪ T) may include sets that are not in σ(S) ∪ σ(T).

4. (a) To show that F is a σ-algebra, we need to show that it is closed under countable unions, countable intersections, and complements. If E is countable, then its complement R \ E is uncountable, and vice versa. Therefore, the complement of any set in F is also in F. If {Ei} is a countable collection of sets in F, then either each Ei is countable, in which case their union is countable, or each complement R \ Ei is countable, in which case the intersection of the complements is countable. Therefore, F is closed under countable unions and intersections, and is a σ-algebra. Since F does not include all Borel sets, it is a proper sub-σ-algebra of the Borel algebra B.

   (b) The σ-algebra generated by the collection {{x} | x ∈ R} is the collection of all countable sets and their complements. This is exactly the set F, so F is the σ-algebra generated by this collection.

5. If A = {A1, A2, . . . , Ak} is a partition of X, then any subset of X can be expressed as a union of some of the Ai. Therefore, the σ-algebra generated by A is the collection of all such unions, which is exactly {⋃j∈J Aj | J ⊆ {1, 2, . . . , k}}.

6. If B(R) is the Borel σ-algebra of R and I = {[a, b) | a, b ∈ R, a ≤ b}, then every open interval (a, b) can be expressed as the union of the intervals in I that are contained in (a, b). Therefore, the σ-algebra generated by I includes all Borel sets, so B(R) = σ(I).

7. A σ-algebra F is a monotone class because it is closed under countable unions and intersections. If (En) is a monotone increasing sequence in F, then the union ⋃∞n=1 En is the limit of this sequence and is in F. Similarly, if (An) is a monotone decreasing sequence in F, then the intersection ⋂∞n=1 An is the limit of this sequence and is in F. Therefore, F is a monotone class.