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課程名稱︰分析二
課程性質︰數學系選修，可抵必修分析導論二
課程教師︰齊震宇
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2017/2/23 - 2017/3/1 13:30
考試時限（分鐘）：如上，不便換算
試題 :
BASIC NOTIONS AND FACTS
1. Abstract disjoint union. Let X (α∈A) be a family of topological spaces.
———————————— α
We define their (abstract) disjoint union (or their coproduct) ∥ X (or
— α
α∈A
∥ X or ∥ X for short) to be the topological space whose underlying set
— α — ‧
α
is ∪ (X ×{α}) as a subset of ( ∪ X ) ×A, and whose topology T
α∈A α α∈A α
consists of subsets of the form ∪ (U ×{α}) where U is an open subset of
α∈A α α
X for every α∈A. (It easily checked that T is a topology.) We have natural
α
ι
α
maps x∈X ——→ (x,α)∈ ∥ X (α∈A), which can be easily seen to be
α — ‧
continuous. Note that ∥ X is the union of disjoint sets ι ( X ) = X ×{α}
— ‧ α α α
(α∈A). We may view ι ( X ) as a copy of X , and hence can view ∥ X as
α α α — ‧
formed by putting all X together "disjointly in a topological manner." It is
α
direct to see that ( ∥ X , {ι } ) has the following universal property:
— ‧ α α
for any set of data ( Y,{ g } ) consists of a topological space Y and a
α α
continuous map g from X to Y for every α, there exists a unique map
α α
g
∥ X ——→ Y such that g。ι = g for every α∈A.
— ‧ α α
Remark. If we replace ∥ X by the usual union ∪ X and replace ι by the
— ‧ ‧ α
natural inclusion map from X to the union, then the universal property will
α
not hold even set theoretically when the sets X have nonempty intersection.
α
2. The σ-algebra generated by a family of subsets. Let Y be a set and S a
————————————————————————
family of subsets of Y. We define the σ-algebra A(S) generated by S to be the
intersection of all σ-algebra on X which contains S. It is direct to see that
for any measurable space (X,M) and any map f from X to Y, f is measurable with
-1
respect to M and A(S) if and only if f (s)∈M for any s∈S.
3. Initial σ-algebras. Given measurable spaces (X ,A ) (j∈J) and a family of
—————————— j j
f
j
maps Z ——→ X (j∈J) from a set Z, a σ-algebra C on Z makes all f s
j j
-1
measurable if and only if C contains {f (E)∣j∈J,E∈A }. Then
j j
-1
A = A := A({f (E)∣E∈A ,j∈J}) is the smallest σ-algrbra on Z
{f.} {f }(j∈J) j j
j
making all f s measurable, which we call the initial σ-algebra on Z induced
j
by the maps f (j∈J).
j
4. Product σ-algebras. If in 3. we take Z to be the Cartesian product set
———————————
Π X and take the maps f to be the natural projection π :(x ) → x ,
j∈J j j j k k∈J j
then we denote the initial σ-algebra A by ☒ A (or ☒ A for short)
{π.} j∈J j ‧
and call it the product σ-algebra of A (j∈J). ( Π X , ☒ A ) is called
j j∈J j j∈J j
the product measurable space of (X ,A )(j∈J). The key property of the product
j j
g
construction is that for any measurable space (Y,M) and any map Y —→ Π X ,
j∈J j
y →(g (y))
j j∈J
the map g is (M,☒ A )-measurable if and only if g is (M,A )-measurable for
‧ j j
every j∈J.
QUESTIONS
1. (gluing process) A set of transition / gluing data
f
jk
T = ( X (j∈J), X ——→ X (j,k∈J) ) (or ( {X } ,{f } for short)
j kj jk j j jk j,k
consists of a family of topological space X (j∈J) and a family of continuous
j
maps f (j,k∈J) such that for any j,k,l∈J the following conditions hold:
jk
(0) X = X , (i) X is an open subset of X , (ii) f is a homeomorphism,
jj j jk j jk
(iii) X ∩X = f (X ∩X ), and (iv) f = f 。f on X ∩X ,i.e., the
jk jl jk kj kl jl jk kl li lk
following diagram commutes:
X ∩X
kj kl
∣ ↖ f
f ∣ ╲ kl
jk ∣ ╲
↓ ╲
X ∩X ←———— X ∩X
jk jl f lj lk
jl
(Following the notation in 1.,) we define a relation ～ on ∥ X as follows:
T — j
j
for any p = (x ,j)∈ι (X ) and q = (x ,k)∈ι (X ), we say that p～q if
j j j k k k T
x =f (x ). It can be shown that ～ is an equivalence relation on ∥ X , whose
j jk k T — ‧
quotient space we denote by X and the natural projection we denote by
T
π : ∥ X ——→ X . Prove the following statements.
T — ‧ T
(1) ～ is an equivalence relation on ∥ X .
T — ‧
(2) The map π 。ι : X —→ V :=π (ι (X )) is a homeomorphism and V is an
T j j j T j j j
open subset of X .
T
(3) (For any map f from a set A to a set B, we define its graph
Γ := {(a,f(a))∣a∈A}.) X is Hausdorff if and only if Γ is a closed
f T f
jk
subset of X ×X (equipped with the product topology) for every pair of indices
k j
j,k∈J.
f p
2. Let X ——→ Y be a quotient map, A a subset of X, and A ——→ f(A) the map
mapping a∈A to f(a)∈f(A).
(1) Show that p is a quotient map if A is a f-saturated open or a f-saturated
closed subset of X.
(2) Give an example to show that p might not be a quotient map without assuming
A to be f-saturated in (1).
(3) Give an example to show that p might not be a quotient map without assuming
A to be either open or closed in (1).
f
3. (For any map X ——→ Y between sets and any subset S⊆Y, we let f to
S
-1 -1
denote the map from f (S) to S which maps x∈f (S) to f(x).)
f
Let X ——→ Y be a map between topological spaces and consider the following
items: (a) continuous map, (b) closed map, (c) open map, (d) homeomorphism,
(e) embedding.
Consider the following two statements.
(1) If f is a ___, then so is f for every open subset W of Y.
W
(2) For any open cover V (j∈J) of Y, if f is a ___ for every j∈J, then so is
j V
j
f.
Answer separately for (1) and (2), which among the items listed above, when put
into ___, will make the statement hold.
4. Let X (j∈J ) be a family of Hausdorff spaces. Let X := Π X be the
j 0 j∈J j
0
product space. For any J⊆J'⊆J we let X := Π X and let π : X —→ X be
0 J j∈J j JJ' J' J
the projection map mapping (x ) to (x ) . (If J' = J we will write
j j∈J' j j∈J 0
π as π ; if J={j} we will write π as π .) A subset of X is called
JJ' J JJ' jJ'
-1
a cylinder with compact base if it is of the form π (Z) for some compact
J
subset Z of X .
J
Now suppose that C (n∈N) is a family of cylinders with compact base with
n
the finite intersection property, i.e., ∩ C ≠ψ for every finite subset
n∈S n
∞
S⊆N. Show that ∩ C ≠ ψ following the instructions below.
n=1 n
(i) For each n∈N choose a finite subset J of J and a compact subset K
n 0 J
n
-1
of X such that C = π (K ).
J n J J
n n n
(ii) Let J := ∪ J . For each j∈J , choose n ∈N such that j∈J ;
* n∈N n * j n
j
besides, let H :=π (C ).
j j n
j
(iii) Choose (a ) ∈X such that a ∈H for every j∈J .
j j∈J j j *
0
/ H if j∈J ;
(iv) Let K:= Π E where E := j *
j∈J j j \ {a } if j∈J \J .
0 j 0 *
(1) Show that K is compact.
(v) Let D:={ F⊆N∣F≠ψ, and F is finite }. Then (D,⊆) is a directed set.
(vi) For every F∈D, let J := ∪ J .
F n∈F n
(2) Show that ( ∩ C )∩( ∩ C )≠ψ for every F∈D.
n∈F n j∈J n
F j
(vii) For every F∈D, select b ∈( ∩ C )∩( ∩ C ). Thus, for every
F n∈F n j∈J n
F j
j∈J we have π (b )∈π (C )=H .
F j F j n j
j
(viii) For every F∈D, define x ∈X by setting its j-th component to be
F
/ π (b ) if j∈J ;
j F F
\ a if j∈J \J .
j 0 F
(3) Check that (x ) is a net in K.
F F∈D
(4) Show that for every n∈N, the net (x ) lies in C eventually.
F F∈D n
∞
(5) Show that ∩ C ≠ψ.
n=1 n
5. Let X (α∈A) be a family of topological spaces. Prove the following
α
statements.
(1) Given a basis S of X for every α∈A,
α α
S := { Π E ∣E ∈S for every α∈A and {α∈A∣E ≠ X } is finite set}
α∈A α α α α α
is a basis of the product space Π X .
α∈A α
(2) (We let B(X) to denote the σ-algebra of all Borel sets of a topological
space X.) We have ☒ B(X ) ⊆ B( Π X ); besides, equality holds if A=N
α∈A α α∈A α
and X is second countable for every α∈A.
α
6. Let (X,A) be a measurable space. We define a relation R on X by requiring
that, for any p,q∈X, pRq if and only if ∀ E∈A [ p∈E ⇔ q∈E ]. It is clear
that R is an equivalence relation. An equivalence class with respect to R is
called an (traditional) atom of (X,A).
(1) Let f be a [0,∞]-valued measurable function on (X,A). Show that f| is a
A
constant if A is an atom of (X,A).
(2) If A=A(S) for some countable family S of subsets of X, then every atom of
(X,A) lies in A.
(3) Let μ be a probability measure on (X,A). Show that "for any ε＞0 there
exists finitely many disjoint measurable sets E ,...,E ,A ,...,A such that
1 m 1 n
μ(E )＜ε for all k, A are atoms, and μ(A )≧ε for all j." is false by
k j j
giving a counterexample.
f
7. Let T ——→ Y be a map between topological spaces. Given τ∈T and L∈Y,
we say that f(t) → L as t →τ if for any (open) neighborhood V of L in Y
there exists an (open) neighborhood U of τ in T such that f(t)∈V for every
t∈U\{τ}. (Note that this definition generalizes the classical definition for
the metric space case.)
(1) It is direcct to see that if f(t) → L as t →τ, then f(t ) → L as
n
n → ∞ for every sequence t in T\{τ} which converges to τ. Show that the
n
converse holds if T is first countable.
f
Let (X,A,μ) be a measure space and T an open subset of R. Let X ×T → R be
a function and τ∈T. Suppose that
(a) f(x,τ) is μ-integrable as a function in x.
(b) for every x∈X fixed, the function f (t):=f(x,t) is differentiable at
x
τ, and
f(x,t)-f(x,τ)
(c) there exists a μ-integrable function g such that∣———————∣
t-τ
≦ g(x) for all (x,t)∈X ×(T\{τ}).
∂f
(2) Show that f(x,t) (for every t∈T) and ———(x,τ) are all μ-integrable
∂t
d ∂f
as a function in x and ——∣ ∫ f(x,t)dμ(x) = ∫ ———(x,τ)dμ(x).
dt t=τ X X ∂t
8. For any probability measure μ on (R,B(R)), we define its characteristic
itx
function (a variant of its Fourier transform) to be ψ (t):= ∫ e μ(dx).
μ R
k k
(1) Show that for any k∈N, ψ is a C function in t∈R if ∫ |x| μ(dx)＜∞;
μ R
(k) k k itx
if this is tha case, we have ψ (t) = i ∫ x e μ(dx)＜∞.
μ R
2
1 -x /2
(2) For any E∈B(R), let μ(E) = ∫ ———— e dx.
E √(2π)
It is direct to verify that μ is a probability measure on (R,B(R)), whose
characteristic function is denoted by ψ(t). Show that ψ'(t)+tψ(t) = 0 and
solve ψ(t).
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