MTL510 Measure Theory
Quiz
23 March 2022
1. State whether the following statement is true or false. Justify.
If f : [0, ∞) → R is a differentiable function, then f ′ is measurable.
2. Let P denote the ternary Cantor set. Define f : [0, 1] → R as follows. If
x ∈ P , then f (x) = 0. Recall that [0, 1] \ P is the disjoint union of open
intervals, the middle thirds that were removed in the construction of P .
Define f to be a constant on each of these open intervals, namely


1
for x ∈ ( 13 , 23 )


for x ∈ ( 19 , 29 ) ∪ ( 79 , 89 )
f (x) = 2


...
In general, f (x) = k on each of the removed middle third open intervals
R
1
of length k in [0, 1] \ P . Is f Lebesgue integrable ? Calculate [0,1] f dm.
3
3. For each n ∈ N, let fn : [0, 1] → R be defined by
nx
fn (x) =
, ∀ x ∈ [0, 1].
1 + n10 x10
(a) Find the pointwise limit f of the sequence (fn )n∈N .
(b) Is this convergence uniform ?
(c) Is the following statement about the convergence of the Riemann
integrals of the above sequence of functions true ? Justify.
Z 1
Z 1
fn (x) dx →
f (x) dx as n → ∞.
0
0
4. Let (fn )n∈N be a monotonic increasing sequence of nonnegative Riemann
integrable functions on [0, 1] such that
lim fn (x) = f (x), ∀ x ∈ [0, 1].
n→∞
State whether the following statement is true or false. Justify.
Z 1
Z 1
f (x) dx = lim
fn (x) dx ?
0
n→∞
0
[4×5 Marks = 20 Marks]
1