TWENTY QUESTIONS FOR THE MIDTERM
1. Assume everything you know from the notes on Riemann-Stieltjes integration. (a) Show that if
g : [0, 1] → R is Riemann-integrable on [0, 1] and ϕ : [a, b] → R is continuous on an interval [a, b] ⊇ g[[0, 1]],
then ϕ ◦ g is Riemann-integrable on [0, 1]. (b) Assume that f : [0, 1] → R is a function for which f 0 (t) exists
R1
at each t ∈ [0, 1] (one-sidedly at endpoints) and that the Riemann integral 0 f 0 (t) dt exists. Show rigorously
that the freshman-calculus formula
s
2
Z 1
df
L[f; 0, 1] =
1+
dt
dt
0
is correct.
2. Assume everything you know from the notes on Riemann-Stieltjes integration. Suppose g and h are
two functions of bounded variation on [a, b] and that V [g; z, x] ≤ V [h; z, x] for each a ≤ z < x ≤ b. Show
Z b
Z b
f(t) dh(t) exists in sense (I), then so does
f(t) dg(t). Is the corresponding statement correct
that if
a
a
for integrals in sense (II)? If you believe not, is there an additional hypothesis that would make it correct?
3. Assume everything you know from the notes on Riemann-Stieltjes integration. (a) Suppose the
Z b
integral
f(t) dg(t) exists (in either sense). Show that if g0 (t) exists at each t ∈ [a, b] and is (bounded and)
a
Z b
Riemann-integrable on [a, b], then that Riemann-Stieltjes integral is equal to
f(t) g0 (t) dt, which exists
a
in the same sense (I) or (II). (b) Show, conversely, that if the latter integral exists (in either sense) then so
does the former (in the same sense).
4. Is the following statement correct? There exists a continuous function g(x, y) ∈ C([0, 1]2, R) on the
unit square with the following property: given any f ∈ C([0, 1], R) and given any > 0, there exist points
n
X
αk g(·, yk ) < . (Here, as usual,
y0 , . . . , yn in [0, 1] and real coefficients α0 , . . . , αn , such that f(·) −
k=0
∞
g(·, yk ) denotes the function x 7→ g(x, yk ).) If you believe it to be correct, exhibit such a function or show
how it may be constructed.
5. [The Riemann-Lebesgue Lemma:] A function f : R → R is said to be of compact support if f ≡ 0
outside some interval [a.b]. Prove that if f is a continuous function of compact support, then
Z
lim
Z
∞
|ω|→∞ −∞
f(t) cos(ωt) dt = 0 and
∞
lim
|ω|→∞
f(t) sin(ωt) dt = 0 .
−∞
(Note that these are Riemann integrals over finite intervals.)
6. Let (X, d) and (Y, ρ) be compact metric spaces. Their Cartesian product X × Y , given the metric
δ((x1 , y1 ), (x2 , y2 )) = d(x1 , x2 ) + ρ(y1 , y2 ), is easily seen to be a (sequentially) compact metric space also.
The set of all functions in C(X × Y, R) of the form
(x, y) 7→
r
X
fk (x) · gk (y), where all fk ∈ C(X, R) and gk ∈ C(Y, R)
k=1
is called C(X) ⊗ C(Y ) (it can in fact be identified with the abstract-algebraic tensor product). Assuming
things that you are supposed to know and checking their hypotheses carefully, show that C(X) ⊗ C(Y ) is a
subalgebra of C(X × Y, R) that is dense in the k · k∞ -metric.
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Math 501:01 Fall 1999
7. (a) Let f : [0, 1] → R be a continuous function, and let G(f) = {(x, y) ∈ R2 : y = f(x), x ∈ [0, 1]}
denote its graph. Can G(f) have positive 2-dimensional Lebesgue measure? (b) Let f : [0, 1] → R2 be a
continuous function, and let Tr(f ) = {y ∈ R2 : y = f (t), t ∈ [0, 1]} denote its carrier (= image set). Can
Tr(f ) have positive 2-dimensional Lebesgue measure? (Explain why or why not. You need not explicitly
construct an example if you say that it can have positive measure.)
8. Suppose you know how to construct Cantor sets of arbitrary measure in [0, 1]. Exhibit (a) a subset
of [0, 1] that is a dense Gδ of measure 0; (b) a subset of [0, 1] that is of first category but measure 1. Can
you exhibit a closed nowhere dense subset of [0, 1] that is of measure 1?
9. If T : Rn → Rn satisfies a Lipschitz condition, then we know that it sends measurable sets to
measurable sets. With the same definition of the Lipschitz condition, is it true that if T : R3 → R2 satisfies
a Lipschitz condition, then it sends (three-dimensionally) Lebesgue measurable sets to (2-dimensionally)
Lebesgue measurable sets?
10. Assume what you know about the effect of nonsingular linear transformations of Rn on ndimensional Lebesgue measure and measurability of subsets of Rn . (a) Let N ⊆ R be a set of Lebesgue
measure zero. Show that N × R ⊆ R2 is a set of 2-dimensional Lebesgue measure zero. (b) Let N ⊆ R be
a set of Lebesgue measure zero. Show that {(x, y) ∈ R2 : x − y ∈ N } is a set of 2-dimensional Lebesgue
measure zero.
11. Let f : [0, 1] → Rn be a continuous path of finite length. Show that if n > 1 then the n-dimensional
Lebesgue measure of its carrier (= image set) Tr(f ) = f [[0, 1]] is zero.
12. It is not difficult to show that the 2-dimensional Lebesgue measure of a Euclidean disc of radius
r > 0 in R2 is πr2 . Suppose E ⊆ R3 is a set and A > 0 is a number, such that for any given > 0, the set E
∞
∞
can be covered by a countable family
P∞ of 2Euclidean spheres {Sk }k=1 all of whose corresponding radii {rk }k=1
are < and for which the sum k=1 πrk ≤ A. Show that the 3-dimensional Lebesgue measure of E is zero.
13. There are situations in which one encounters disjoint families A = {Eα }α∈A of measurable subsets
of Rn with 0 < m(Eα ) < ∞ for each α ∈ A that are maximal with respect to those properties, i.e., there
is no properly larger disjoint family of measurable sets of finiteSpositive measure that contains A. Show (a)
that such a family A is necessarily countable; (b) that Rn = [ α∈A Eα ] ∪ N , where m(N ) = 0. (Use what
you know about Lebesgue-measurable sets.)
14. Let f : [0, 1] → (−∞, +∞] be a lower-semicontinuous function. Give a definition of l. s. c., and
using your definition, show that if f 6≡ +∞, then there is a point x0 ∈ [0, 1] at which f attains a minimum
(and finite) value.
15. Let f : [0, 1] → (−∞, +∞] be a lower-semicontinuous function. Give a definition of l. s. c., and
using your definition, show that there is an increasing sequence {gk }∞
k=1 of continuous real-valued functions
dominated by f and such that g1 (x) ≤ g2 (x) ≤ · · · ≤ gk (x) % f(x) at each x ∈ [0, 1].
16. Let f(x, y) be a bounded R-valued function defined on [0, 1] × [0, 1]. Suppose f(x, y) is separately
continuous, i.e., that x 7→ f(x, y) is continuous for fixed y and y 7→ f(x, y) is continuous for fixed x.
(a) Prove that there is a sequence of (jointly) continuous functions on [0, 1] × [0, 1]—say {fk (x, y)}∞
k=1 —
bounded by the same bounds as f(x, y), which converges pointwise to f(x, y) on [0, 1] × [0, 1]. (b) Deduce
that such a separately continuous function is jointly continuous somewhere. (Can you say more?) (c)
Deduce that separately continuous functions on [0, 1] × [0, 1], bounded or not, are measurable (with respect
to 2-dimensional Lebesgue measure).
n
∞
17. Let {fk }∞
k=1 be a sequence of continuous real-valued functions defined on R . Suppose that {fk }k=1
n
n
is pointwise bounded on R , that is, for each x ∈ R there exists Mx ≥ 0 for which |fk (x)| ≤ Mx holds for
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Math 501:01 Fall 1999
all k ∈ N. Show that there is an open set U ⊆ Rn with U = Rn , such that for every compact K ⊆ U there
exists a (finite) MK such that |fk (x)| ≤ MK uniformly for all k ∈ N and x ∈ K.
n
18. Let {fk }∞
k=1 be a sequence of measurable real-valued functions defined on a measurable set E ⊆ R
∞
of finite measure 0 < m(E) < ∞. Suppose that {fk }k=1 is pointwise bounded on E, that is, for each x ∈ E
there exists Mx ≥ 0 for which |fk (x)| ≤ Mx holds for all k ∈ N. Show that given any > 0 there exists a
compact K ⊆ E with m(E \ K) < , and a finite M , such that |fk (x)| ≤ M uniformly for all k ∈ N and
x ∈ K.
19. Give an example of a sequence of measurable real-valued functions (a) that converges to zero
pointwise on R but does not converge to zero in measure; (b) that converges to zero in measure but does
not converge to zero pointwise almost everywhere.
n
20. Prove the theorem (of F. Riesz): if {fk }∞
k=1 is a sequence of functions on a measurable set E ⊆ R
∞
∞
and {fk }k=1 converges in measure to f : E → R, then f is measurable and there is a subsequence {fkj }j=1
with the property that: for every > 0 there is a measurable F ⊆ E with m(E \ F ) < , such that
|f(x) − fkj (x)| → 0 uniformly for x ∈ F .
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Math 501:01 Fall 1999