Real Analysis Syllabus

Real Analysis Syllabus
The real analysis prelim will be based on the material related to the topics listed below. This list
is not meant to be exhaustive, but is intended to be a guide to subjects to be studied thoroughly.
1. Metric Spaces: Completeness, compactness, connectedness, Ascoli-Arzela Theorem,
Baire Category Theorem, Stone-Weierstrass Theorem.
2. Measure and Integration: Integration theory on general measure spaces including
Lebesgue integral and Lebesgue-Stiel jes integral on the line, Lusin’s Theorem, Egoroff’s
Theorem, Fatou’s Lemma, the Monotone and Dominated Convergence Theorems, convergence in measure, dense subspaces of L1 , e.g., simple functions, continuous functions with
compact support..., Radon-Nikodym Theorem, Tonelli and Fubini Theorems.
3. Lp spaces: Hoelder and Minkowski Inequalities, Riesz Representation Theorem for Lp
spaces, completeness of Lp , dense subsets of Lp .
4. Differentiation: Differentiation of monotone functions, differentiation of an indefinite
integral, functions of bounded variation, absolute continuity, Fundamental Theorem of
Lebesgue Calculus, Lebesgue points.
References:
G. Folland, Real Analysis
W. Rudin, Real and Complex Analysis
H.L. Royden, Real Analysis (2nd Ed.)
C. Apostol, Mathematical Analysis (2nd Ed.)
1
Real Analysis Prelim
January 2008
1. Let (X, B, µ) be a measure space. Suppose f and g are non-negative integrable functions
such that A ∈ B implies
Z
Z
gdµ.
f dµ =
A
A
Prove that f = g a.e.
2. Suppose T = {z ∈ C : |z| = 1}. Suppose f ∈ L1 (T) = ”‘the absolutely Lebesgue integrable
functions on T.”’ If
Z π
1
ˆ
f (n) =
f (x)e−int dx,
2π π
prove that
lim fˆ(n) = 0.
|n|→∞
3. Let A ⊂ [0, 1] be a measurable set of positive measure. Prove that there exist x, y ∈ A
such that x 6= y and x − y is rational.
R1
R1
4. Let f, g be two nonnegative measurable functions on [0, 1] such that 0 f (x)dx = 1, 0 g(x)dx =
R1
2, 0 (g(x))2 dx = 5. Let E = {x|g(x) ≥ f (x)}. Show that m(E) ≥ 51 where m is Lebesgue
measure.
5. Set
Z
∞
h(x) =
(x + y)−1 f (y)dy
0
L1 (0, ∞).
for f ∈
for all a > 0.
Show that h is
C∞
away from the origin. Show as well that h0 ∈ L1 [a, ∞)
6. Let f : [0, 1] 7→ R be absolutely continuous and satisfy f (0) = 0 as well as
Z
1
|f 0 (x)|2 dx < ∞.
0
Show that
1
lim x− 2 f (x)
x↓0
exists and find the limit.
2
Real Analysis Prelim
August 2007
1. Let {f
} be a sequence of positive Lebesgue measurable functions defined on [0, 1] such
Pn∞
that n=1 m({x ∈ [0, 1] : fn (x) > 1}) < ∞. Show that
lim sup fn (x) ≤ 1
n→∞
for almost all x ∈ [0, 1]. Here m denotes Lebesgue measure on [0, 1].
2. Let f ∈ Lp [0, 1], where 2 < p ≤ ∞. Show that
Z
|f (x)|
√ dx < ∞.
x
[0,1]
3. If 1 < p < ∞, let {fn } be a sequence of elements of Lp [0, 1] such that there exists M > 0
with ||fn ||p ≤ M for all n > 0. Suppose that the sequence {fn } converges pointwise almost
everywhere to a function f defined on [0, 1].
(a) Prove that f ∈ Lp [0, 1].
(b) Let q be the conjugate exponenet to p. Prove that for each function g ∈ Lq [0, 1],
Z
Z
lim
fn (x)g(x)dx =
f (x)g(x)dx.
n→∞ [0,1]
[0,1]
[Hint: you might find Egoroff’s Theorem to be helpful.]
4. Let [a, b] and [c, d] be two closed and bounded intervals in R.
(a) If f1 is defined on [c, d] and satisfies a Lipschitz condition there, and f2 is a function
that is absolutely continuous on [a, b] whose range is contained in [c, d], show that
the composite function f1 ◦ f2 is absolutely continous on [a, b]. (Recall a function
f : [c, d] → R is said to satisfy a Lipschitz condition on [c, d] if there exists M > 0
such that |f (x) − f (y)| ≤ M |x − y|, ∀x, y ∈ [c, d].)
(b) Let f : [a, b] → R be a function. For each of the following statements, either prove it
is correct, or show it is false by providing a counterexample.
i. If f is of bounded variation over [a, b], then f is absolutely continuous on [a, b].
ii. If f is absolutely continuous on [a, b], then f is of bounded variation over [a, b].
5. Let f : [0, 1] × [0, 1] → R be a bounded function, and suppose that for each fixed t ∈ [0, 1]
the function f (·, t) is a Lebesgue measurable function of x. For each (x, t) ∈ [0, 1] × [0, 1]
∂f
suppose the partial derivative ∂f
∂t exists. Suppose also that ∂t is bounded over [0, 1]×[0, 1].
Prove that for all t0 ∈ [0, 1]
Z 1
Z 1
d
∂f
[
f (x, t)dx]t=t0 =
(x, t0 )dx.
dt 0
0 ∂t
6. Let f be a non-negative Lebesgue integrable function defined on R, and let m2 denote
two-dimensional Lebesgue measure on R2 .
(a) Prove that the sets {(x, y) : 0 < y < f (x)} and {(x, y) : 0 ≤ y ≤ f (x)} are measurable
with respect to the Lebesgue product measure on R2 , and establish the equalities
Z
m2 ({(x, y) : 0 < y < f (x)}) = m2 ({(x, y) : 0 ≤ y ≤ f (x)}) =
f (x)dx.
R
3
(b) Define φ : [0, ∞) → [0, ∞] by φ(t) = m({x ∈ R : f (x) ≥ t}). Prove that φ is a
decreasing function of t and that
Z
Z
f (x)dx.
φ(t)dt =
[0,∞)
R
4
Real Analysis Prelim
January 2006
p
1. For 1 ≤ p < ∞, write
the classical Banach space of all Lebesgue-integrable functions
R ∞ L for
on R that satisfy −∞ |f |p < ∞, and where the norm is defined by
Z
∞
||f ||p =
p
|f | dx
1
p
.
−∞
(a) Suppose a sequence {fn } of elements of Lp converges to an element f of Lp , i.e.,
limn→∞ ||f − fn ||p = 0. Prove that limn→∞ ||fn ||p = ||f ||p .
(b) Suppose p and q are numbers strictly between 1 and ∞, and assume that p1 + 1q = 1.
Suppose a sequence {fn } of elements of Lp converges to an element f of Lp , and
that a sequence
{gn } of elements Rof Lq converges to an element g of Lq . Prove that
R∞
∞
limn→∞ −∞ fn gn dx converges to −∞ f gdx.
2. For each t ∈ R and f ∈ L1 , where L1 is as defined in Question 1, define
Ut (f )(x) = f (x − t), x ∈ R.
Fix f ∈ L1 . Prove that the map t → Ut (f ) is uniformly continuous as a map from R to
L1 .
3. Prove that
Z 1
∞
X
1
1
=
dx.
n
x
n
0 x
n=1
R∞
You may use the Gamma function evaluation 0 e−t tk dt = k!.
4. If f is a real-valued function on R, write Sf0 for the set of all x such that f is differentiable
at x. Call a set S a differentiation set if there exists a function f for which S = Sf0 .
(a) Show that Sf0 +g contains Sf0 ∩Sg0 . Show by example that it can happen that Sf0 +g = R
but Sf0 ∩ Sg0 = ∅.
(b) Show that every open interval I is a differentiation set.
(c) Let E be a Lebesgue-measurable subset of R with m(E) < ∞. Show that, for each
> 0, there exists a differentiation set S for which m(E∆S) < , where m denotes
Lebesgue measure, and E∆S is the symmetric difference of the sets E and S.
5. Let f and g be real-valued Lebesgue measurable functions on [0, 1], not assumed to be
integrable. Let E = {(x, y) ∈ [0, 1] × [0, 1] : f (x) = g(y)}.
(a) Prove that E is measurable with respect to the Lebesgue product measure m × m
defined on [0, 1] × [0, 1].
(b) Suppose in addition that m × m(E) = 1. Prove that there is a real constant c such
that f ≡ g ≡ c, m a.e. on [0, 1].
6. A sequence of functions {fn } ∈ L1 [0, 1] is said to be uniformly integrable if
Z
lim sup
|fn (x)|dx = 0.
c→∞ n
{x∈[0,1]:|fn (x)|<c}
Show that for such a sequence, if fn → f almost everywhere for some f , then fn → f in
L1 [0, 1].
5
R
Step 1: First show that supn [0,1] |fn |dx < ∞.
Step 2: Next show thatR the (almost-everywhere) limit f is indeed in L1 [0, 1].
Step 3: Last, estimate [0,1] |f − fn |dx, employing Egorov’s Theorem to finish the job.
6
Real Analysis Prelim
August 2006
All problems are worth the same number of points.
Clearly explain the reasoning necessary to achieve your answers.
1. Let {fn } be a sequence of real-valued measurable functions defined on the real line. Let
E = {x : limn→∞ fn (x) exists}. Show E is a measurable set.
2. Suppose (X, S, µ) is a measure space. Let g and h be real-valued measurable functions
defined on X. Let {fn } be a sequence of real-valued measurable functions defined on X.
Suppose that for all n, g ≤ fn ≤ h, µ-almost everywhere.
R∞
3. Let f ∈ L1 (R). Show limn→∞ −∞ f (x) sin(nx)dx = 0.
4. Let f be a real-valued measurable function defined on [a, b]. Then given > 0 show there
exists a continuous function g on [a, b] such that µ{x : f (x) 6= g(x)} < . Here µ is
Lebesgue measure.
5. Let f and g be real-valued absolutely continuous functions defined on [0, 1].
(a) Show that the product f g is also absolutely continous on [0, 1].
R1
R1
(b) If g(0) = f (1) = 0 then show 0 f 0 (x)g(x)dx = − 0 f (x)g 0 (x)dx. Also, explain why
these integrals exist.
R1
R1
(c) Suppose f and h are in L1 (R). Suppose furthur that 0 h(x)g(x)dx = − 0 f (x)g 0 (x)dx
for all absolutely continous functions g with g 0 ∈ L∞ [0, 1] and g(0) = 0. Show f is
absolutely continous and h = f 0 , µ-almost everywhere.
Here µ is Lebesgue measure.
6. Prove the following. Let (X, S, µ) and (Y, T ν) be σ-finite measure spaces. Suppose K(x, y)
is a real-valued measurable function with respect to µ × ν. Suppose there exists a C > 0
such that
Z
|K(x, y)|dν ≤ C, for all x ∈ X
Y
Z
|K(x, y)|dµ ≤ C, for all y ∈ Y .
X
RFor all f ∈ Lp (Y, T, ν), 1 < p < ∞ the function T (f ) defined on X by T (f )(x) =
Y K(x, y)dν exists µ-almost everywhere. Show ||T (f )||p ≤ C||f ||p where ||f ||p is the
Lp (Y, T, ν) norm of f .
7
Real Analysis Prelim
August 2005
1. Show that, if f ∈ L1 (R), then
Z
lim
δ→0 R
|f (x + δ) − f (x)|dx = 0.
You may use the fact that the space CC (R) of continuous, Rcomplactly supported functions
on R is dense in L1 (R), with respect to the norm ||g||1 = R |g(x)|dx.
2. For this problem, you may take it as given that, if f and g are measurable functions on R,
then F (x, y) = f (x − y)g(y) defines a measurableR function on R2 . The convolution f ∗ g
of measurable functions f, g on R is the function f ∗ g, of a real variable x, defined by
Z
f (x − y)g(y)dy.
f ∗ g(x) −
R
The domain of f ∗ g is the set of all x such that f ∗ g(x) is defined (that is, such that
F (x, y) = f (x − y)g(y) defines a Lebesgue integrable function of y).
(a) Show that, if f, g are continuous on R and vanish outside R+ = (0, ∞) then
Z x
f ∗ g(x) =
f (x − y)g(y)dy,
0
and then use this to show that f ∗ g is continuous and vanishes outside R+ . Hint for
continuity: Note that
Z x0
Z x
f ∗ g(x) − f ∗ g(x0 ) =
(f (x − y) − f (x0 − y))g(y)dy +
f (x − y)g(y)dy.
0
x0
(b) Show that, if f, g ∈ L1 (R), then f ∗ g ∈ L1 (R) (in particular, f ∗ g(x) is defined for
almost all x ∈ R. , and ||f ∗ g|| − 1 ≤ ||f ||1 ||g||1 .
3. For this problem, you may assume that for any function f on R to R the set of all points
of discontinuity is an Fσ set.
For the following three questions, if your answer is yes, explicitly construct such an example. If your answer is no, give an argument that no such function exists.
(a) Does there exist a function from R to R that is continuous at exactly one point?
(b) Does there exist a function from R to R that is continuous at every irrational point
and discontinuous at every rational point?
(c) Does there exist a function from R to R that is continuous at every rational point,
and discontinuous at every irrational point?
4. (a) Prove Tchebyschev’s inequality, which says the, if f is nonnegative and integrable on
[0, 1] and µ is Lebesgue measure on [0, 1], then
Z
1
µ({x ∈ [0, 1] : f (x) ≥ c}) ≤
f (x)dx.
c [0,1]
8
(b) We define an ultraCauchy sequence in a metric space (X, d) to be a sequence a1 , a2 , ...
in X such that, for some c > 0,
d(ak , ak+1 ) ≤
c
k3
for all k ∈ Z+ .
Let h1 , h2 , ... be an ultraCauchy sequence in L2 (0, 1), the space of square-integrable
functions on [0, 1], with metric
sZ
|f (x) − g(x)|2 dx.
d(f, g) = ||f − g||2 =
[0,1]
Show that the hk ’s converge pointwise almost everywhere: that is, show that limk→∞ hk (x)
exists for almost all x ∈ [0, 1].
Hint: For each k ≥ 1, define Gk = {x ∈ [0, 1] : |hk (x) − hk+1 (x)|2 ≥ k14 }. Use part
(a) to show that, given > 0, there is an N ∈ Z+ such that HN = ∪∞
k=N Gk has a
measure less than . Then show that, if x ∈
/ ∩N HN , {hk (x)} is a Cauchy sequence.
9
Real Analysis Prelim
January 2004
1. Let λ be Lebesgue measure on [0, 1] and let φ : [0, 1] → [a, b] be measurable. Define a set
function µ on the Borel sets of [a, b] by µ(A) = λ(φ−1 (A)).
(a) Prove that µ is a measure on the Borel sets of [a, b].
(b) If f : [a, b] → R is a bounded measurable function, probe that
Z
Z
f (φ(x))dλ(x) =
f (x)dµ(x).
[0,1]
[a,b]
2. Let f : R → [0, ∞) be an integrable function such that
Z
|x|f (x)dx < ∞.
R
Prove that
d
dt
R
R cos(tx)f (x)dx
=−
R
R sin(tx)xf (x)dx.
3. Let f and g be bounded continuous functions on [0, ∞). Define
t
Z
f ∗ g(t) =
f (t − x)g(x)dx, t ≥ 0.
0
If φ is a measurable function on [0, ∞) define
Z ∞
L(φ)(s) =
e−sx φ(x)dx
0
R∞
for all s > 0 for which 0 e−sx |φ(x)|dx < ∞.
Prove that for all s > 0, L(f ∗ g)(s) = L(f )(s)L(g)(s).
4. Let f : R → R be a Lebesgue measurable function that is an element of L1 (R). For n > 0
set
An = {x ∈ R : |f (x)| > n}.
(a) Prove that
lim λ(An ) = 0,
n→∞
where λ is Lebesgue measure on R.
(b) Prove that
lim nλ(An ) = 0.
n→∞
5. (a) Suppose f is an integrable function on R. Suppose further that f is absolutely
continuous on each closed and bounded interval [a, b], and that its derivative f 0 also
is integrable over R. Prove that limx→∞ f (x) = 0.
(b) Give an example of an integrable, absolutely continuous function f ∈ L1 (R) for which
limx→∞ f (x) does not exist.
10
Real Analysis Prelim
August 2004
1. Let (X, B, µ) be a measure space. Suppose f and g are non-negative integrable functions
such that A ∈ B implies
Z
Z
gdµ.
f dµ =
A
A
Prove that f = g a.e.
2. Suppose T= {z ∈ C : |z| = 1}. Suppose f ∈ L1 (T)=”The absolutely Lebesgue integrable
functions on T.” If
Z π
1
ˆ
f (n) =
f (x)e−int dx,
2π −π
prove that
lim fˆ(n) = 0.
|n|→∞
3. Let A ⊂ [0, 1] be a measurable set of positive measure. Prove that there exist x, y ∈ A
such that x 6= y and x − y is rational.
R1
4. Let f, g be two nonnegative measurable functions on [0, 1] such that 0 f (x)dx = 1,
R1
R1
1
2
0 g(x)dx = 2, 0 (g(x)) dx = 5. Let E = {x|g(x) ≥ f (x)}. Show that m(E) ≥ 5
where m is Lebesgue measure.
5. Set
Z
∞
h(x) =
(x + y)−1 f (y)dy
0
L1 (0, ∞).
for f ∈
for all a > 0.
Show that h is
C∞
away from the origin. Show as well that h0 ∈ L1 [a, ∞)
6. Let f : [0, 1] → R be absolutely continuous and satisfy f (0) = 0 as well as
Z
1
|f 0 (x)|2 dx < ∞.
0
Show that
1
lim x− 2 f (x)
x↓0
exists and find the limit.
11
Real Analysis Prelim
January 2002
1. Let (X, ρ) be a separable metric space. Suppose S ⊆ X. Show that there exists a countable
set F ⊆ S such that F is dense in S, i.e., S ⊆ F = closure of F .
2. Let (X, S, µ) be a measurable space for which µ(X) < ∞ and E ∈ S implies µ(E) ≥ 0. Let
f : X → R be S-measurable. Suppose g : R → R+ = {x ∈ R : x ≥ 0} is Borel measurable.
For a Borel set E ⊆ R, define ν(E) = µ(f −1 (E).
(a) Show that ν is a measure.
(b) Show that
Z
Z
g ◦ f dµ.
gdν =
X
R
3. Suppose that f : R → R has a derivative at every point of R. Is it always true that f 0 is
bounded on [−1, 1]? It it is, give a proof. If not, give a counterexample.
4. Let f : [0, ∞) → R be a bounded measurable function such that limx→∞ x2 f (x) = 1. Find
an integral expression for
R∞
(1 − cos(x))f ( λx )dx
lim 0
.
λ2
λ→0+
Justify your answer.
5. Let {fn } and f be measurable functions on [0, 1] with Lebesgue measure. Suppose that
fn converges in measure to f , and that for some p > 1, supn ||fn ||p < ∞.
(a) Prove that fn converges to f in L1 .
(b) Find a counterexample to the statement in part (a) if p = 1.
6. Let (X, S, µ) be a measure space such that µ(X) > 0. Recall that L∞ (X, S, µ) is a metric
space under the L-infinity norm. An element E of S is called an atom if µ(E) > 0 and
whenever F ∈ S satisfies F ⊆ E, then either µ(F ) = 0 or µ(E \ F ) = 0.
(a) Suppose there is a sequence E1 , E2 , ... in S such that m 6= n implies Em ∩ En = ∅
and µ(En ) is always strictly positive. Show that L∞ (X, S, µ) is not separable.
(b) Show that if L∞ (X, S, µ) is separable, there there are finitely many atoms, E1 , ..., En
in S, such that
µ (X \ ∪∞
k=1 ) = 0.
12
Real Analysis Prelim
August 2002
1. (16 pts) Let (X, d) be a metric space, and let A ⊆ X. Recall that a point x ∈ X is said to
be an accumulation point of the set A if every neighborhood of x contains a point a ∈ A
with a 6= x.
(a) Let x ∈ X. Prove that the set {x} is nowhere dense in X if and only if x is an
accumulation point of X.
(b) State, but do not prove, the Baire Category Theorem.
(c) Prove that if (X, d) is complete and each x ∈ X is an accumulation point of X, then
X is uncountable.
(d) Using part (iii), deduce that R is uncountable.
2. (17 pts)
(a) Define simple functions, taking their domain to be a bounded interval [a, b] in R.
(b) Show that sums and products of simple functions are simple.
(c) Define step functions, taking their domain to be a bounded interval [a, b] in R.
(d) Let f be a bounded measurable function on a bounded interval [a, b]. Define the
Lebesgue integral of f on [a, b].
(e) Let f be a bounded measurable function on a bounded interval [a, b], which is Riemann
integrable. Show that the Riemann and Lebesgue integrals of f on [a, b] coincide.
3. (16 pts) Let E be a Lebesgue-measurable subset of R, and assume that the measure of E
is 0. Let AE be the subset of R2 consisting of the points (x, y) for which x2 + y ∈ E.
(a) Assuming that AE is a measurable subset of R2 , compute its measure.
(b) Prove that AE is a Lebesgue-measurable subset of R2 .
4. (17 pts) Let f ∈ L1 ([0, 1]), and suppose that f1 , f2 , ... is a sequence of absolutely continuous
functions on [0, 1] such that f (x) = limn→∞ fn (x) for every x ∈ [0, 1].
(a) Suppose the sequence f10 , f20 , ... converges in L1 to a function g. Prove that f is
absolutely continuous.
(b) Suppose that there exists an L1 function g for which |fn0 (x)| ≤ g(x) for every x ∈ [0, 1]
and for every n ∈ N. Prove that f is absolutely continuous.
(c) Show by giving an example, that the L1 norms of the functions fn0 can be uniformly
bounded, and yet f is not absolutely continuous.
5. (9 pts) Prove that it is impossible to find bounded sequences of numbers, a1 , a2 , ... and
b1 , b2 , ... such that limn→∞ [an cos(nx) + bn sin(nx)] = 1a.e. on [0, 2π].
6. (16 pts) Prove Egoroff’s Theorem, as stated here (m denotes Lebesgue measure): Let
f1 , f2 , ... be a sequence of measurable functions from [0, 2π] to R that converges a.e. to a
real-valued measurable function f . Then for every > 0, there is a subset A ⊂ [0, 2π] with
m(A ) < , such that f1 , f2 , ... converges uniformly to f on [0, 2π]\A .
7. (9 pts) Prove the following form of the Riemann-Lebesgue Lemma. If f ∈ L1 (R), then
Z
lim
f (x) cos(nx)dx = 0
n→∞ R
13
and
Z
lim
n→∞ R
f (x) sin(nx)dx = 0.
Hint: As a first step, prove the statements for the case in which f is the characteristic function
of a finite interval.
14
Real Analysis Prelim
January 2001
1. Let m be Lebesgue measure on [0,1] and let {fn } be a sequence of measurable functions
on [0,1]. Either prove or give a counter-example to each of the following statements:
a) If fn converges to f a.e. (m) then fn converges to f in measure (m).
b) If fn converges to f in measure (m) then fn converges to f a.e. (m).
2. Let α ∈ R and set f (x) = x sin( x1α ) if x > 0 and f (0) = 0. For which α is f of bounded
total variation on [0,1]? Prove that your answer is correct.
3. Let m be Lebesgue measure on R, and let {fn } be a sequence of functions in L2 (m) with
||fn ||2 ≤ 1 for all n.
a) If fn converges to f in measure (m) prove that f ∈ L2 (m) and that ||f ||2 ≤ 1.
b) If in addition ||fn ||2 → ||f ||2 prove that fn → f in L2 (m).
4. Let (X, B, µ) be a finite measure space and f : X → [0, ∞) be aR B measurable function.
Let A be a sub σ-algebra of B and define ν : A → R by ν(E) = E f dµ.
a) Prove that ν is a measure on A.
b) Prove that there is an A measurable function g on X such that for all E ∈ A
Z
ν(E) =
g dµ.
E
5. Let X = Y = [0, 1] and let B be the Borel sets in [0,1]. Let (X, B, µ) be Lebesgue measure
on B and let
measure onR B. Let D = {(x, y) ∈ X × Y : x = y}.
R R(Y, B, ν) be counting
R R
Prove that ( χD dµ) dν, ( χD dν) dµ, and χD d(µ × ν) are all unequal. Here χD is
the indicator (characteristic) function of the set D.
6.
a) Give an example of a measure space (X0 , B0 , µ) for which 1 ≤ p < q ≤ ∞ implies that
Lp (µ) ⊂ Lq (µ). (No proof required)
b) Give an example of a measure space (X1 , B1 , ν) for which 1 ≤ p < q ≤ ∞ implies that
Lq (ν) ⊂ Lp (ν). (No proof required)
c) Give an example of a measure space (X2 , B2 , λ) for which for any 1 ≤ p < q ≤ ∞ both
Lp (λ) 6⊆ Lq (λ) and Lq (λ) 6⊆ Lp (λ). (No proof required)
d) Choose one of your examples in a), b), or c) and prove that it is an example having
the claimed property.
15
Real Analysis Prelim
August 2001
1. Let f1 , f2 , ... be a sequence of continuous real-valued functions on a metric space (X, d)
and suppose that the sequence converges uniformly to a function f on X. Prove that f is
continuous. Is this result still true if the sequence is only assumed to converge pointwise
to f ? If so give a proof; if not give a counter-example and explain it.
2. Let (X, d) be a metric space and let x1 , x2 , ... be a Cauchy sequence in X. Prove that if
the sequence has a cluster point, x, then it converges to x.
3. A function f : [0, 1] → R is said to be Holder continuous of order 1/2 provided there exists
1
a constant C such that |f (x) − f (y)| ≤ |x − y| 2 for all x and y in [0, 1]. Let f1 , f2 , ... be
a sequence of Holder continuous functions of order 1/2, taking [0,1] to [0,1], such that the
same constant C suffices for all of them. Show that there is a subsequence of f1 , f2 , ... that
converges uniformly on [0,1] to a continuous function f .
4. A function f : [0, 1] → R is said to satisfy a Lipschitz condition provided there is a constant
M such that |f (x) − f (y)| ≤ M |x − y| for all x, y in [0,1].
(a) Show that a function f satisfying a Lipschitz condition on [0,1] is absolutely continuous.
(b) Show that an absolutely continuous function f satisfies a Lipschitz condition iff |f 0 |
is bounded.
5. Let f : R → R be continuous and of period 2π, i.e. f (x + 2π) = f (x) for all x. Show that
for every > 0, there is a finite Fourier series
φ(x) = a0 +
N
X
(an cos(nx) + bn sin(nx))
n=1
such that |φ(x) − f (x)| < for all x.
6. Let f ∈ L1 (R) and suppose that g : R → R has compact support and is infinitely differentiable. Define h : R → R by
Z
f (t)g(x − t)dt,
h(x) =
R
for x ∈ R. Show that this makes sense and that the resulting function is everywhere
differentiable, with
Z
0
h (x) = f (x)g 0 (x − t)dt
for x ∈ R.
7. Let f, g : R → R be Borel measurable functions such that f (x) = g(y) for almost all
pairs (x, y) relative to Lebesgue measure in the plane. Prove that f and g agree almost
everywhere with constant functions.
16
Real Analysis Prelim
August 2000
1. Suppose f ∈ L∞ [0, 1].
(a) Show that f ∈ Lp [0, 1] for all 1 ≤ p < ∞.
(b) Show that limp→∞ ||f ||p = ||f ||∞ .
2. Suppose f is a measurable
real-valued function defined on R. Suppose for all pairs of real
Rb
numbers a < b, 0 ≤ a f (x)dx ≤ b − a. Show that 0 ≤ f (x) ≤ 1 for almost all x.
3. Suppose f and g are positive
measurable
on [0,1] such that f (x)g(x) ≥ 1 for all x. Prove
R
R1
1
that 1 ≤ 0 f (x) dx
0 g(x) dx .
4. Let f be a continuous function on [0, ∞). Suppose limx→∞ f (x) exists (and is finite).
Prove that f is uniformly continuous on [0, ∞).
5. A function f is continuously differentiable on (0,1). Let fn (x) = n[f (x + n1 ) − f (x)] and let
0 < a < b < 1. Show that fn (x) converges uniformly on (a, b). Find out the limit function.
6. Suppose fn ,gn are non-negative finite functions on [0,1] related to each other by the (nonlinear) integral equation
Z
fn (x) +
1
k(x, y)fn2 (y) dy = gn (x),0 ≤ x ≤ 1,
0
where k(x, y) is a non-negative continuous function on [0, 1] × [0, 1]. Suppose fn are monotone decreasing in n (and so are gn ). Put f = limn→∞ fn , g = limn→∞ gn . Prove that
Z
f (x) +
1
k(x, y)f 2 (y) dy = g(x).
0
17
Real Analysis Prelim
January 6, 1999
Each problem is worth 14 points.
1. If f (x) is differentiable on R, prove that f 0 (x) is Borel measurable.
2. Let X be a set equipped with a σ-algebra A. If mi is a measure on (X, A) for each i, such
∞
X
1
mi (E) for all E ∈ A. Prove
that mi (X) ≤ 1. Define m : A → [0, +∞) by m(E) =
2i
i=1
that m is a measure on (X, A).
3. Compute
Z
lim
n→∞ 0
1
1
1
(1 + nt )n t n
dt.
Justify your answer
4. If f (x) is a Lipschitz function on R, i.e. there exists M such that |f (x) − f (y)| ≤ M |x − y|
for all x, y ∈ R, prove that f 0 (x) exists almost everywhere.
5. Let I = [0, 1]. Prove that the set of all polynomials with rational coefficients is dense in
Lp (I), 1 ≤ p < ∞.
R
6. Suppose that I = [0, 1], 0 < p < 1, p1 + 1q = 1. |f (x)|p dx < ∞, g(x) 6= 0 for all x ∈ I and
Z
1 Z
1
Z
p
q
R
p
q
q
|g(x)| dx < ∞. Prove |f (x)g(x)|dx ≥
|f (x)| dx
|g(x)| dx .
I
I
I
7. Assume that f is Lebesgue integrable on [0, 2π]. Prove that
π
lim
n→∞ 2
Z
2π
Z
f (x)| sin(nx)|dx =
0
f (x)dx.
0
18
2π
Real Analysis Prelim
1. If 1 < p < ∞ and 1 < q < ∞ such that
show that fn gn → f g in L1 .
1
p
+
August 1999
1
q
= 1 and fn → f in Lp , gn → g in Lq , then
2. Assume f is continuous on R. Put
n−1
fn (x) =
1X
k
f (x + )
n
n
k=0
Prove that fn (x) converges uniformly on every closed finite interval [a, b]. Find the limiting
function.
3. If A ⊂ R2 is measurable and every vertical line meets A in a countable set, what is the
Lebesgue measure of A?
4.
(a) Give an example of a sequence of integrable functions fn on R that are continuous
and converge pointwise to an integrable function f and are such that
Z
Z
fn 6→ f.
(b) Let fn be a sequence of nonnegative measurable functions on [0, 1] that converge to
f . If f is integrable and fn ≤ f + 1 ∀n show that
Z 1
Z 1
f = lim
fn .
0
0
R∞
5. Let f be a positive continuous function such that −∞ f (x) dx is finite. Assume that, for
some T > 0, f is decreasing on [T, ∞) and it is increasing on (−∞, T ]. Show that
F (x) =
∞
X
f (x + k)
k=−∞
is a continuous periodic function with the period 1.
19
6. Let φn , n = 1,2,. . . be a complete orthonormal system in the space L2 (R, λ), where λ
stands for Lebesgue measure.
(a) Show that, for every Borel set B ⊂ R of strictly positive Lebesgue measure, one has
1≤
Z X
∞
B
|φn (x)|2 dx.
1
Hint: Assuming λ(B) < ∞, apply the Parceval identity to 1B . Then use Schwartz’s
inequality.
P
2
(b) Show that ∞
1 |φn (x)| = ∞ a.e.
7. Every real number x ∈ [0, 1) has the unique binary representation
x=
∞
X
sn (x)2−n
n=1
where sn (x) = 0 or 1 and sn (x) = 0 for infinitely many n.
(a) Show that sn (x) are Borel measurable functions.
(b) Evaluate
Z
1
Z
(s1 + · · · + sn ) dxand
0
0
20
1
(s1 + · · · + sn )2 dx.