Departments of Mathematics
Fall 2014
Montana State University
Prof. Kevin Wildrick
Measure Theory
Problem Set 3
Due Friday, September 18th , 11:00 am.
The symbol (?) indicates that this problem can not be found directly in Bartle’s book, and must
be solved and turned in. Other problems should be solved but need not be turned in.
1. (?) A mapping f : R → R is L-Lipschitz if for any pair of points x, y ∈ R,
|f (x) − f (y)| ≤ L|x − y|.
Show that there is a constant C ≥ 1, depending only on L, such that for any subset E ⊆ R
and any L-Lipschitz function f : R → R,
m∗ (f (E)) ≤ Cm∗ (E).
Find the smallest possible value of C for a given L. Bonus: solve this problem in Rn .
2. (?) Suppose that f : (0, 1) → (0, 1) is differentiable at every point. Show that if E ⊆ R
satisfies m(E) = 0, then m(f (E)) = 0 as well. Hint: Consider, for each integer k ≥ 1, the
set
Ik = {x0 ∈ R : if |x − x0 | < 1/k, then |f (x) − f (x0 )| < k|x − x0 |.}.
Show that f |Ik is a Lipschitz function, and use a variant of Problem 1.
3. (?) Let {fk : [0, 1] → R}k∈N be a sequence of mappings with the following properties
• for each > 0, there is a positive integer N so that for any x ∈ [0, 1] and any k ≥ N ,
|fk (x) − fN (x)| < .
In other words, we assume that the sequence {fk } is a uniform Cauchy sequence.
• for each > 0 and each x ∈ [0, 1], there is a number δ > 0 so that for any k ∈ N and
y ∈ [0, 1],
|x − y| < δ =⇒ |fk (x) − fk (y)| ≤ .
In other words, we assume that the sequence {fk } is equicontinuous.
Show that f : [0, 1] → R given by
f (x) = lim fk (x)
k→∞
defines a continuous function. Bonus: show that f is in fact uniformly continuous.
4. Show that a set E ⊆ Rn is measurable if and only if there is a Gδ set G ⊆ Rn and measure
0 set Z ⊆ G so that E ∪ Z = G.