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

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 : A → B be a function and E, F are subsets of A. Show thatf (E ∪ F ) = f (E) ∪ f (F ) and f (E ∩ F ) ⊂ f (E) ∩ f (F )