Mathematics 2224: Lebesgue integral
Tutorial exercise sheet 2
1. Let Σ be a σ-algebra, let µ1 : Σ → [0, ∞] and µ2 : Σ → [0, ∞] be two measures and
let α1 , α2 ∈ [0, ∞]. Show that the map
µ : Σ → [0, ∞],
µ(E) = α1 µ1 (E) + α2 µ2 (E) for E ∈ Σ
is also a measure. [We usually write µ = α1 µ1 + α2 µ2 .]
2. Fix x ∈ R and let Σ be a σ-algebra of subsets of R. Show that the map
(
1
if x ∈ E
δx : Σ → [0, ∞], δx (E) =
for E ∈ Σ
0
if x ∈
6 E
is a measure.
If µ = ∞δ0 + 4δ1 , compute µ(∅), µ({0}), µ([0, 1]), µ((0, 1]), µ((0, 1)) and µ(R).
3. True or false? (And why?)
(a) B is the σ-algebra generated by {(−a, b) : a, b ∈ R, a, b ≥ 0}
(b) B is the σ-algebra generated by {(−a, a) : a ∈ R, a ≥ 0}.
4. (a) Show that B is translation-invariant: if E ∈ B and α ∈ R then α + E ∈ B.
(b) Show there is no translation-invariant probability measure µ : B → [0, ∞].