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）︰2017/6/20 考試時限（分鐘）：180分鐘 試題 : 分析二期末考 06/20/2017 1. 以下七大題請挑選兩大題作答，超過者僅取得分最高的兩大題計分。 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. 以下兩題請挑選一題作答，超過者僅取最高的一題計分。 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.把下面這題作了吧! 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.以下三題請挑選一題作答，超過者僅取得分最高的一題計分。 _ 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.以下兩題請挑選一題作答，超過者僅取得分最高的一題計分。 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 (不限作答題數) 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 --

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