Let μ denote the Lebesgue measure on &mathbb;R and μ* be the associated Lebesgue outer measure. Let A be a non-empty subset of [0, 1]. Which of the following statements is necessarily true?

The counting measure on a finite set X assigns which value to a subset A ⊆ X?

For any sequence of sets, the outer measure m satisfies which of the following properties?

Let μ denote the Lebesgue measure on ℝ. Suppose that f is a non-negative Lebesgue measurable function on ℝ. Let 0 = a0 < a1 < a2 < … be an unbounded sequence such that an+1 ≤ c an for some real number c and for all n ≤ 1. Let Ak = {x ∞ ℝ | ak ≤ f(x) < ak+1} for each k ≤ 0. Which of the following statements is true?