Real Analysis qual study guide

Exercise 1.2. If A,B ⊂ R,m∗(A) = 0, then m∗(A ∪B) = m∗(B) Proof: m∗(A ∪B) ≤ m∗(A) +m∗(B), and m∗(B) ≤ m∗(A ∪B), hence we have m∗(B) ≤ m∗(A ∪B) ≤ m∗(A) +m∗(B) = m∗(B) ∴ m∗(A ∪B) = m∗(B) Exercise 1.3. Prove E ∈M iff ∀ > 0,∃O ⊂ R open, such that E ⊂ O and m∗(O\E) < Proof: (⇒) O\E = E ∩O implies that m∗(O\E) = m∗(Ec ∩O), but we have m∗(O) = m∗(Ec ∩O) +m∗(E ∩O) So suppose m∗(E) < ∞ ⇒ m∗(Ec ∩ O) = m∗(O)−m∗(E ∩ O). Let In be a countable cover for E, so In = (an, bn). Let On = (an, bn + 2−n ) and let O = ⋃ On. Then m∗(O) = ∑ l(On) = ∑ 2−n + bn − an = + ∑ bn − an, and m∗(E ∩O) = m∗(E) since E ⊂ O. So we have m∗(E ∩O) = m∗(E) ≤ ∑ l(In) = ∑ bn − an ⇒ m∗(E ∩O) ≤ ∑ l(On)− ∑ l(In)