Question

Let f be measurable and B a Borel set. Then f-1[B] is a measurable set. [Hint: The class of sets for which f-1[E] is measurable is a σ-algebra.

Answer 1

Proof. Let = { E ⊂ : f-1[E] is Lebesgue measurable}. Since inverse images are well behaved with respect to all the usual set operations and the family of Lebesgue measurable sets is a σ- algebra, it is easy to see that is a σ-algebra. Here are the details of that. Suppose E ε . Then f-1[E] is Lebesgue measurable. Consequently, ~f-1[E] = f-1[~E] is Lebesgue measurable.

Hence, ~E ε . Similarly, if {Ej} is a countable family of elements of E, then {f-1[Ej]} is a countable family of Lebesgue measurable sets. Consequently, ∪f-1[Ej]=f-1[∪Ej] is Lebesgue measurable. Hence ∪Ej ε . Since, f-1[∅] = ∅,

we now know that is a σ-algebra. Since the definition of measurability implies that contains each open ray, (a,∞), must contain the open intervals, and hence, all open subsets of the real line. Thus, must contain the σ-algebra generated by the open sets, the Borel sets.

Let = { E ⊂ : f-1[E] is Lebesgue measurable}. Since inverse images are well behaved with respect to all the usual set operations and the family of Lebesgue measurable sets is a σ- algebra, it is easy to see that is a σ-algebra. Here are the details of that. Suppose E ε . Then f-1[E] is Lebesgue measurable. Consequently, ~f-1[E] = f-1[~E] is Lebesgue measurable.

Hence, ~E ε . Similarly, if {Ej} is a countable family of elements of E, then {f-1[Ej]} is a countable family of Lebesgue measurable sets. Consequently, ∪f-1[Ej]=f-1[∪Ej] is Lebesgue measurable.

Hence ∪Ej ε . Since, f-1[∅] = ∅, we now know that is a σ-algebra. Since the definition of measurability implies that contains each open ray, (a,∞), must contain the open intervals, and hence, all open subsets of the real line. Thus, must contain the σ-algebra generated by the open sets, the Borel sets