臺灣大學數學系

九十一學年度第一學期碩博士班資格考試題

分析(Analysis)

Sept 11, 2002

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Points distributions: A(40), B(30), C(15), D(15)

A.

Let $[a,b]$ be a closed interval of ${\Bbb R}$. Define the spaces
$B[a,b] = \{ f \mid f:[a,b] \rightarrow {\Bbb R} \mbox{ is bounded } \}$
$BV[a,b] = \{ f \mid f:[a,b] \rightarrow {\Bbb R} \mbox{ is a function of bounded variation} \}$
For $f \in B[a,b]$, denote $\parallel f \parallel _{\infty}=\sup_{x \in [a,b]} \mid f(x) \mid$ to be the sup-norm of $f$.

(a)

Prove that $B[a,b]$ with the sup-norm is a Banach space, and $BV[a,b] \subset B[a,b]$. Must $BV[a,b]$ be a closed subspace of $B[a,b]$?

(b)

Prove that every $f \in BV[a,b]$ is Lebesgue measurable, and $BV[a,b] \subset L^{1}([a,b])$. Must every $f \in BV[a,b]$ be Riemann integrable? If yes, is the Riemann integral of $f$ equal to the Legesgue integral of $f$?

(c)

Let $f_{n} \in BV[a,b]$ $(n=1,2,3, \cdots)$ be a sequence, and $\lim_{n \rightarrow \infty} f_{n}=f \in BV[a,b]$ uniformly. Let $\Gamma _{n}$ and Γ be the arclength of the graphs of $f_{n}$ and $f$ respectively. Must $\lim_{n \rightarrow \infty} \Gamma _{n}=\Gamma$ be true?

(d)

Must $B[a,b] \subset \cap _{p \ge 1} L^{p}([a,b])$? If no, what is $\cap _{p \ge 1} L^{p}([a,b])$?

B.

Determine which of the following statements are true or false. Prove your assertion.

(a)

Let $A \subset {\Bbb R}^{n}$ and $B \subset {\Bbb R}^{m}$. Then, even when $A \times B$ is Legesgue measurable, $A$ and $B$ are not necessarily Legesgue measurable. But if $A \times B$ is Borel measurable, then $A$ and $B$ must be Borel measurable.

(b)

Let $E_{m} \subset {\Bbb R}^{n}$ be a sequence of Lebesgue measurable sets, then

$$\lim_{n \rightarrow \infty} \mid A - \cup^{m}_{j=1}E_{j} \mid _{e}=\mid A - \cup^{\infty}_{j=1}E_{j} \mid _{e}$$

holds for any $A \subset {\Bbb R}^{n}$ with $\mid A \mid _{e} < \infty$. The notation $\mid A \mid _{e}$ denotes the Lebesgue outer measure of $A$.

(c)

Let $E \subset {\Bbb R}^{n}$ be Lebesgue measurable and $f_{m} \in L^{1}(E)$. Assume that $f_{1} \ge f_{2} \ge \cdots \ge f_{m} \cdots \ge 0$ a.e. in $E$, and $f_{m} \rightarrow 0$ a.e. in $E$, then $\sum^{\infty}_{m=1}(-1)^{m-1}f_{m} \in L^{1}(E)$, and

$$\int _{E} (\sum^{\infty}_{m=1}(-1)^{m-1}f_{m}(x))dx=\sum^{\infty}_{m=1}(-1)^{m-1} \int_{E}f_{m}(x)dx.$$

C.

Let $E \subset {\Bbb R}^{n}$ be Lebesgue measurable and $f_{m}$ be a sequence of measurable functions in $E$ ocnverging a.e. to $f$. Assume that there exists another measurable sequence $g_{m}$ in $E$ such that $0 \le \mid f_{m}(x) \mid \le g_{m}(x)$ a.e. in $E$ for all $m$, and $g_{m}$ converges in $L^{1}(E)$. Prove that $\lim_{m \rightarrow \infty} \int_{E} f_{m}(x) dx=\int_{E}f(x)dx$.

D.

Let $f \in L^{p}(E) \cap L^{q}(E)$ with $1 \le p \le q \le \infty$ where $E$ is a Lebesgue measurable set.

(a)

Prove that $\parallel f \parallel _{r} \le \parallel f \parallel ^{\alpha}_{p} \parallel f \parallel ^{1- \alpha}_{q}$ for all $p \le r \le q$, where $\frac{1}{r}=\frac{\alpha}{p}+\frac{1- \alpha}{q}$.

(b)

prove that, for all $\epsilon > 0$, $\parallel f \parallel _{r} \le \epsilon \parallel f \parallel _{p}+ \epsilon^{- \mu} \parallel f \parallel _{q}$, where $\mu = \frac{\alpha}{1- \alpha}$.

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