臺灣大學數學系

八十七學年度第一學期碩博士班資格考試試題

分析(加考)

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A.

Choose 4 from the following 6 problems

1.

(a)

Let $\{ f_n \}$ be a sequence of measurable functions on $[0,1]$. Assume $\lim_{n \rightarrow \infty}f_n(x)=f(x)$ for all $x \in [0,1]$. Show that $f$ is measurable.

(b)

Let $f$ and $g$ be two measurable functions. Show that $fg$ is measurable.

2.

Assume $f \in L^p({\mathbb{R}}^n)$ with $1 \leq p < \infty$. (a) Let $z \in {\mathbb{R}}^n$ and $f_{z}(x)=f(x+z)$. Show that

$$\vert\vert f_{z}-f\vert\vert _{L^p} \rightarrow 0 \,\,\hbox{as} \,\,z \rightarrow 0.$$

(b) Let

$$g(y)=\int_{{\mathbb{R}}^n}f(x+y)e^{-\vert x\vert^2-\vert y\vert^2} \,dx.$$

Is $g$ continuous? Is $g$ differentiable?

3.

Let $f,f_n\in L^p([0,1])$ with $1 \leq p < \infty$. Show that if $f_n \rightarrow f$ a.e. and $\vert\vert f_n\vert\vert _{L^p}\to\vert\vert f\vert\vert _{L^p}$ as $n\to\infty$, then $\vert\vert f_n-f\vert\vert _{L^p}\rightarrow 0$ as $n \rightarrow \infty$.

4.

Let $f$ be of bounded variation on $[a,b]$.
(a) Show that ${df \over dx}$ exists a.e. on $[a,b]$.
(b) Let $V[a,b]$ be the variation of $f$. Show that if

$$V[a,b]=\int_a^b \vert{df \over dx}\vert\,dx,$$

then $f$ is absolutely continuous on $[a,b]$.

5.

Find the value of $p$ for which the function $f(x,y)=x^{-{1 \over 3}}(x+y)^{-{1 \over 5}}$ is in $L^p([0,1]^2)$.

6.

Let $f \in C^1([0,1])$ and $Z=\{x\in[0,1]: {df \over dx}(x)=0\}$. Show that the set $f(Z)=\{f(x):x\in Z\}$ has measure zero.

B.

Choose 1 from the following 2 problems

1.

Show that the image of an open set under a nonconstant analytic function is an open set.

2.

Evaluate the intrgral

$$\int_0^{\infty}{dx \over{1+x^n}} \ ,$$

where $n\geq 2$ is a positive integer.

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