Math 410 Midterm Exam
Instructions: Due Thursday April 14. You may use your class
notes and the textbook, but no other resources are allowed. Do not
discuss the problems with anyone else.
(1) (17 points) Let f : Rn → Rm be a continuous function. Prove
that the set B = {(x, y) ∈ Rn+m : y = f (x)} has measure
zero in Rn+m .
(2) (17 points) Prove that every closed subset of Rn which has
measure zero is rectifiable.
(3) (17 points) Exhibit a bounded closed subset of R which is not
rectifiable.
(4) (17 points) Let A be a bounded open subset of Rn . Let f :
A → R be a uniformly continuous function (i.e., for every > 0
there exists δ > 0 such that if x, y ∈ A satisfy kx − yk∞ < δ
then |f (x) − f (y)| < ). Prove that f is integrable over A in
the extended sense.
(5) (32 points) Let S be a subset of Rn . Given x ∈ Rn , define the
set
S − x = {y − x : y ∈ S}.
(a) Assume that S is rectifiable and let x ∈ Rn . Prove that the
set S − x is also rectifiable and that vol(S − x) = vol(S).
(b) Let Q0 and Q1 be rectangles in Rn . Prove that the function
F : Rn → R defined by
F (x) = vol (Q0 − x) ∩ Q1
is continuous.
(c) Let Q0 , Q1 , and F be as in part (b). Prove that the set
Q2 = {z − y : z ∈ Q0 and y ∈ Q1 }
is a rectangle in Rn and that F (x) = 0 whenever x 6∈ Q2 .
(d) Let Q0 , Q1 , and F be as in part (b). Prove that F is
integrable over Rn and that
Z
F = vol(Q0 ) · vol(Q1 )
Rn
(Hint: use Fubini’s Theorem along with parts (a), (b), (c))
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