Mathematics 2224: Lebesgue integral
Tutorial exercise sheet 1
1. Let a ∈ R. Show that (a, ∞) =
µ((a, ∞)) = ∞.
S∞
j=1 (a, a
+ j], and deduce that (a, ∞) ∈ L and
2. Let a, b ∈ R with a ≤ b. Show that [a, b] ∈ L, and µ([a, b]) = b − a.
3. Find sets E1 , E2 ∈ L which show that the rule µ(E1 ) = µ(E2 ) =⇒ µ(E1c ) = µ(E2c )
is false.
4. Let A, B ∈ L with B ⊆ A.
(a) Prove that if µ(A) < ∞, then µ(A \ B) = µ(A) − µ(B).
(b) Give a counterexample to show that if µ(A) = ∞, then this rule may fail.
Explain what goes wrong with the proof of (a) if you try to apply it to your
counterexample.