课程名称︰分析二
课程性质︰数学系选修，可抵必修分析导论二
课程教师︰齐震宇
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2017/6/20
考试时限（分钟）：180分钟
试题 :
分析二期末考 06/20/2017
1. 以下七大题请挑选两大题作答，超过者仅取得分最高的两大题计分。
Λ
Setting for 1.1-1.4. Let X be a locally compact Hausdorff space and C (X)—→C
c
be a C-linear map which is positive in the sense that it maps C (X) into
c ≧0
[0,∞).
1.1.(20 points) Under the above setting,
(1) complete the statement of Riesz's representation theorem,
(2) write down the definitions of the outer measure μ, A and A in the proof
F
of Riesz's representation theorem given in the lecture, and
(3) show that μ is countably subadditive.
1.2.(20 points) Under the setting of X, Λ, μ, A and A in the proof of
F
Riesz's representation theorem given in the lecture, show that
(1) for every compact K ⊆ X we have K ∈ A and μ(K) = inf{Λf | K ﹤f},
F
(2) for every open V ⊆ X we have μ(V) = sup{μ(K) | K ⊆ V, K is compact},
and
(3) μ(∪ A ) = Σ μ(A ) for every disjoint family A ∈ A ( j ∈ N );
j j j j j F
if furthermore μ(∪ A )＜∞, then ∪ A ∈ A .
j j j j F
1.3.(20 points) Under the setting of X, Λ, μ, A and A in the proof of
F
Riesz's representation theorem given in the lecture, show that
(1) for any A ∈ A and ε＞0, there exist a compact K and an open V such that
F
K ⊆ A ⊆ V ⊆ X and μ(V﹨K)＜ε,
(2) A ﹨A , A ∪A , and A ∩A ∈ A for all A , A ∈ A , and
1 2 1 2 1 2 F 1 2 F
(3) A is a σ-algebra containing B .
X
1.4.(25 points) Under the setting of X, Λ, μ, A and A in the proof of
F
Riesz's representation theorem given in the lecture, show that
(1) A = { S ∈ A | μ(S)＜∞ } and (2) Λf = ∫ fdμ for all f ∈ C (X).
F X c
1.5.(25 points) Let (X,A,μ) be a σ-finite measure space. Show that the map
T
q T p f
L (μ) ————→ L (μ)* : f → ( g —→ ∫ gfdμ )
X
is an isomorphism preserving the corresponding norms if 1/p＋1/q＝1 and
1≦p＜∞.
1.6.(25 points) State and prove the Radon-Nikodym theorem and Lebesgue's
decomposition theorem.
k
1.7.(35 points) Let μ be a finite positive measure on (R , B ). We define
R^k
the maximal function of μ (Mμ)(x) := sup {(Q μ)(x) | 0＜r＜∞} where
r
μ(B_r(x)) k
(Q μ)(x) := ——————— for every x ∈ R .
r λ(B_r(x))
(1) Show that Mμ is a lower semicontinuous.
(2) (Assume the simple version of Vitali's covering lemma.) Show that
k
k 3
λ({x ∈ R | (Mμ)(x)＞c})≦ ——∥μ∥.
c
1 k
(3) Let f ∈ L (R ). Write down the definition of a Lebesgue point of f and
k
Show that almost every x ∈ R is a Lebesgue point of f.
2. 以下两题请挑选一题作答，超过者仅取最高的一题计分。
A
2.1.(10 points) Let I be an open interval and I ——→ M (R) and
n
b n
I ——→ M (R) ～ R be continuous matrix valued functions. (In other
n×1 ＝
words, A(t) = (a (t)) and b(t) = b (t) where a and b are continuous
jk j jk j
n
R-valued functions on I.) Given t ∈ I and x ∈ R , show that if φ is any
0 0 0
n n
R -valued continuous function on I, then the sequence of R -valued functions
t
φ (t) := x ＋∫ (A(s)φ (s)＋b(s))ds (t ∈ I)
n 0 t_0 n-1
converges compactly in I.
2.2.(20 points) Let (X,A)——→(Y,B) be a measurable map. For any positive
－1
measure μ on (X,A) we define (T μ)(B) := μ(T (B)) for every B ∈ B. It
*
is direct to see that T μ is a positive measure on (Y,B), called the push-
*
_
forward of μ via T. Show that, for any R-valued B-measurable function f on Y,
∫ fd(T μ) exists if and only if ∫ f。Tdμ exists, and they are equal if
Y * X
they exist.
3.把下面这题作了吧!
n
3.1.(1)(7 points) Let K be a compact set in R and U an open neighborhood of K.
∞
Show that there exist a sequence of functions ρ ∈ C (U) such that ρ ↘χ .
n c n K
(Hint. You might want to choose a suitable sequence of open sets U containing
n
K and create ρ according the inter-relations between these U .)
n n
n
(2)(8 points) For any open set U in R , we define
f f is Lebesgue measurable and
L (U) := { U —→ C | }.
loc ∫ |f|dλ ＜∞ for all compact K ⊆ U
K n
∞
Let f ∈ L (U) be such that ∫ fρdλ = 0 for all ρ ∈ C (U). Show that
loc U c
f(x) = 0 for almost every x ∈ U.
n
(Hint. Show that ∫ fdλ = 0 for every Lebesgue measurable A in R . Start with
A
the case A be compact.)
4.以下三题请挑选一题作答，超过者仅取得分最高的一题计分。
_
4.1.(10 points) Let (X,A,μ) be a measure space and f an A-measurable R-valued
function such that ∫ fdμ exists (but not necessarily finite). We have known
X
that ν(E) := ∫ fdμ (E ∈ A) is a signed measure on (X,A). Show that for
E
＋ ＋ － －
E ∈ A we have ν (E) = ∫ f dμ, and ν (E) = ∫ f dμ.
E E
1
4.2.(15 points) Let (X,A,μ) be a measure space and g ∈ L (μ) (i.e., a
C-valued integrable function). We have known that ν(E) := ∫ gdμ (E ∈ A)
E
is a complex measure on (X,A). Show that for E ∈ A we have |ν|(E)=∫ |g|dμ.
E
4.3.(15 points) Let (X,A) be a measurable space and μ a complex measure on it.
Then μ＜＜|μ|, and hence, by the Radon-Nikodym theorem, there exists
1
h ∈ L (|μ|) such that μ(E) = ∫ hd|μ| for every E ∈ A.
E
Show that |h| = 1 |μ|-almost everywhere on X.
(Hint. Show both |h|≦1 and |h|≧1 |μ|-almost everywhere on X. (1) Recall the
lemma "averge vs image." (2) For r＞0 let A = {x ∈ X | |h(x)|＜r }. Show that
r
|μ|(A )≦r|μ|(A ).)
r r
5.以下两题请挑选一题作答，超过者仅取得分最高的一题计分。
5.1.(10 points) Let (X,A,μ) be a measure space with μ(X)＜∞ and f a C-valued
A-measurable function. Show that for any 1≦p≦p'＜∞ we have
1 p 1/p 1 p' 1/p'
( ——— ∫ |f| dμ ) ≦ ( ——— ∫ |f| dμ ) .
μ(X) X μ(X) X
(Hint Use Holder's inequality for suitable functions and exponents.)
5.2.(15 points) Let (X,A,μ) be a measure space and f a C-valued A-measurable
function on X. We deine the essential supremum of f
ess.sup|f| := inf { C＞0 | { x ∈ X | |f(x)|＞C } is μ-null }.
Show that if limsup ∥f∥ ＜∞, then ess.sup|f| ≦ limsup ∥f∥ .
p→∞ p p→∞ p
6. Bonus questions (不限作答题数)
6.1.(15 points) Let z ( n ∈ N ) be a sequence in C converging to some z ∈ C.
n
z
n n z
Show that lim (1 ＋ ———) = e according to the following steps: first
n→∞ n
expand the left hand side before taking limit via the binomial theorem. Then,
interpret the expanded result as the integral of a suitable function h with
n
respect to a suitable measure, which is independent of n. Finally, apply
suitable convergence theorem to reach the expected conclusion.
6.2.(15 points) Let (X,A) be a measurable space, μ a finite positive measure
on (X,A), and f a sequence of R-valued measurable functions which converges
n
in measure to an R-valued measurable function f. For any t ∈ R we let
F (t) := μ({ x ∈ X | f (x)≦t }) (n ∈ N) and F(t) := μ({ x ∈ X | f(x)≦t})
n n
Show that, if F is continuous at some t ∈ R, then lim F (t ) = F(t ).
0 n→∞ n 0 0