Mathematics 2224: Lebesgue integral
Tutorial exercise sheet 3
1. Let X be a set and let E, F ⊆ X.
True or false? For the false ones: when are they true, and can you correct them?
(a) χE∩F = min(χE , χF ) = χE · χF
(b) χE∪F = max(χE , χF ) = χE + χF
(c) χE c = 1 − χE
(d) χE\F = χE − χF
[These should all be interpreted pointwise. For example, the first statement means:
χE∩F (x) = min(χE (x), χF (x)) = χE (x) · χF (x) for every x ∈ X.]
2. For the given sequence (an )n≥1 , find lim sup an and lim inf an , and lim an , if the
n→∞
n→∞
n→∞
limit exists.
(a) an = n(−1)n
(b) an = 2 +
n
n+1
n
(c) an = 2(−1) +
n
n+1
n
(d) an = ∞ · ((−1) + 1) + (−1)n
3. Directly from the definition of a Lebesgue measurable function, show that if c ∈ R,
then the constant function f : R → R given by f (x) = c is Lebesgue measurable.
4. (a) Show that if U is an open subset of R and f : U → R is continuous, then the
function
(
f (x) if x ∈ U
g : R → R, g(x) =
0
if x 6∈ U .
is Lebesgue measurable.
(b) Deduce that the function
(
h : R → R,
h(x) =
1
x
0
if x 6= 0
if x = 0
is Lebesgue measurable.
(c) What happens in (a) if U ∈ L but U is not necessarily open?