台灣大學數學系

九十二學年度第二學期博士班資格考試題

實分析

May 9, 2004

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There are Problems A to D with a total of 110 points.Please write down your proof or computatioinal steps clearly on the answer sheets.

A.

Let $(X, \mathcal{M}, \mu)$ be a measure space where $\mathcal{M}$ is a $\sigma$-algebra of subsets of $X$, and $\mu: \mathcal{M} \rightarrow [0, \infty]$ is a measure. Define

$$\mu^*(E) = \inf \{ \mu(A) \vert A \in \mathcal{M}$$   , and $$E \subset A \}$$    , for any subset $$E \subset X$$

(a)

(10 points) Prove that $\mu^*$ is an outer measure define in $X$ such that every $E \in \mathcal{M}$ is $\mu^*$-measurable, and $\mu^*(E) = \mu(E)$. Are sets in $\mathcal{M}$ the only $\mu^*$-measurable subset of $X$?

(b)

(15 points) Let $E_n \subset X(n = 1, 2, 3, \cdots)$ be a sequence of subsets. Probe that $\mu^* (\lim \inf_{n \rightarrow \infty} E_n) \leq \lim \inf_{n \rightarrow \infty} \mu^* (E_n)$. Give example to show that the equality is in general false. Is $\mu^*(\lim \sup_{n \rightarrow \infty} E_n) \geq \lim \sup_{n \rightarrow \infty} \mu^*(E_n)$ also true?

B.

(15 points)Let $f \in L^1 (\mathbb{R}^n)$ and $K\subset \mathbb{R}^n$ be a compact subset of $\mathbb{R}^n$. Define the functions

$$\phi (x)= \int_\mathbb{R}^n \vert f(x+y)+f(y)\vert dy, \bigskip\psi(x)=\int_{k+x} f(y)dy\bigskip$$    for $$\bigskip x \in \mathbb{R}^n$$

where $K+x=\{y+x \vert y \in K \}.$ Prove that $\phi$ and $\psi$ are both continuous functions in $\mathbb{R}^n$.Are both uniformly continuous in $\mathbb{R}^n$?

C.

Let $(X, \mathcal{M}, \mu)$ be a measure space ( as in Problem A). Let $f_n( n = 1, 2, 3, \cdots )$ and $f$ be in $L^p(X, \mathcal{M}, \mu)$ with $1 \leq p < \infty$.

(a)

(15 points) If $\lim_{n \rightarrow \infty} \vert\vert f_n\vert\vert _{L^p} = \vert\vert f \vert\vert _{L^p}$, prove that $f_n \rightarrow f$ in measure iff $\lim_{n \rightarrow \infty} \vert\vert f_n - f \vert\vert _{L^p} = 0$

(b)

(15 points) If $\vert\vert f_n\vert\vert _{L^p}$ is bounded in $L^p$ and $f_n \rightarrow f$ in measure where $1 < p < \infty$, prove that

$$\lim\limits_{n \rightarrow \infty} \int_X f_n(x) g(x) d \mu(x) = \int_X f(x) g(x) d\mu(x)$$    for any $$g \in L^q(X, \mathcal{M}, \mu)$$

where $q$ is given by $\frac{1}{p} + \frac{1}{q} = 1$. Show that by an example that the conclusion is false in case $p = 1$.

D.

Determine which of the following statements is true or false. Prove your answer. Each has 10 points.

(a)

Let $f:[a,b] \rightarrow \mathbb{R}$ be a function of bounded variation. Then $f$ is absolutely continuous in $[a,b]$ iff $f$ is absolutely continuous in $[a+\varepsilon, b]$ for all sufficiently small $\varepsilon > 0$.

(b)

Let $E \subset \mathbb{R}^2$ be a Borel subset, then the set $F = \{ \sin(x^2 + y^2) \vert (x,y) \in E \}$ is also Borel in $\mathbb{R}$. However, if $E$ is Lebesgue measurable, the $F$ may not be Lebesgue measurable.

(c)

There exists a collection $\mathcal{F}$ of closed rectangles in $\mathbb{R}^n$ such that $\cup_{E \in \mathcal{F}} E$ is not Lebesgue measurable.

(d)

Let $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ be Borel measurable. Then, for each $y \in \mathbb{R}$ fixed, the function $f(x,y)$ is also Borel measurable in $x \in \mathbb{R}$.

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