Mathematics 2224: Lebesgue integral
Homework exercise sheet 2
Due 3:50pm, Wednesday 16th February 2011
Recall that m∗ denotes Lebesgue outer-measure, L is the collection of Lebesgue
measurable subsets of R, and µ is Lebesgue measure on R.
1. If E, F ⊆ R, show that m∗ (E ∪ F ) ≤ m∗ (E) + m∗ (F ).
2. If N ⊆ R and m∗ (N ) = 0, prove each of the following statements.
(a)
(b)
(c)
(d)
m∗ (A ∩ N ) = 0 for any A ⊆ R
m∗ (A ∩ N c ) = m∗ (A) for any A ⊆ R
N is Lebesgue measurable
m∗ (A ∪ N ) = m∗ (A) for any A ⊆ R
3. If a, b ∈ R with a ≤ b, show that
m∗ ([a, b]) = m∗ ((a, b]) = m∗ ([a, b)) = m∗ ((a, b)) = b − a.
4. Compute m∗ (R \ Q) and m∗ ([0, 1] \ Q).
5. If α ∈ R and E ⊆ R, recall that α + E = {α + x : x ∈ E}. Show that if E ∈ L,
then α + E ∈ L. [Hint: show that A ∩ (α + E) = α + (−α + A) ∩ E and
(α + E)c = α + E c for any A ⊆ R.]
6. If α ∈ R and E ⊆ R, let αE = {αx : x ∈ E}.
(a)
(b)
(c)
(d)
Show that if α > 0, then m∗ (αE) = αm∗ (E).
Show that m∗ (−E) = m∗ (E), where −E = {−x : x ∈ E}.
Deduce that m∗ (αE) = |α|m∗ (E) for any α ∈ R.
Show that if E ∈ L, then αE ∈ L. [Treat the case α = 0 separately.]
7. Let E0 = [0, 1] and for n ∈ N, let
En = 0.1En−1 ∪ (0.5 + 0.1En−1 )
where 0.1En−1 = {0.1x : x ∈ En−1 }.
(a) Compute the sets E1 and E2 .
(b) Prove by induction on n that for every n ∈ N0 :
(i). En is the disjoint union of 2n closed subintervals of [0, 1], each of Lebesgue
measure 10−n , and hence En ∈ L and µ(En ) = 5−n .
P
−j
(ii). Writing 0.d1 d2 d3 . . . = ∞
j=1 dj · 10 , we have
En = {0.d1 d2 d3 . . . : dj ∈ {0, 5} if 1 ≤ j ≤ n, and dj ∈ {0, 1, 2, . . . , 9} for j > n}.
∞
T
(c) Let E =
Ej .
j=0
(i). Prove that E ∈ L and µ(E) = 0.
(ii). Explain why
E = {0.d1 d2 d3 . . . : dj ∈ {0, 5} for all j ≥ 1},
and use Cantor’s diagonal argument to show that E is uncountable.