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

Let f be a function defined on a set X with values in a set Y . For any subsetE of Y , letf −1(E) = {x ∈ X | f (x) ∈ E}.Show that the setFX = {f −1(E) | E ∈ FY }is a σ-algebra of subsets of X if FY is a σ-algebra of subsets of Y

Let (X, F) be a measurable space and let f be a function from X to Y . Let Abe a collection of subsets of Y such that f −1(E) ∈ F for every E ∈ A. Showthat f −1(D) ∈ F for every set D which belongs to the σ-algebra generated byA.

Let X be a set and let {Fi | i ∈ I} be an arbitrary collection of σ- algebras. Showthat the collection F := {F | ∀i ∈ I, F ∈ Fi} is a σ-algebra.

Which of the following collections F of sets (in X) are σ-algebras? Which ones arealgebras? Explain each answer.(a) X = {1, 2, 3, 4}, F = {∅, {1}, {2}, {3, 4}, {1, 2}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}}.(b) X = {1, 2, 3, ...} and F = {A ⊆ X | either A or X \ A is finite}.(c) X is an uncountable set and F = {A ⊆ X | either A or X \ A is countable}.(d) X is any set, F1 ⊆ F2 ⊆ F3 ⊆ · · · , are σ-algebras in X and F = ⋃∞n=1 Fn

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.