臺灣大學數學系

 八十九學年度第二學期碩博士班資格考試試題

 分析

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一、

(a)

Give an example which shows that the image of a measurable set under a continuous transformation may not be measurable.

(b)

If $f$ is absolutely continuous on $[a,b]$, show that the image under $f$ of any measurable subset of $[a,b]$ is measurable.

二、

Let $f$ be a measurable real-valued function on $[a,b]$. Show that given $\delta > 0$, there is a continuous function $g$ on $[a,b]$ such that the Lebesgue measure $\vert\{ x : g(x) \neq f(x) \}\vert < \delta$. Can you do the same on the interval $(-\infty, \infty)$? (Lusin's Theorem)

三、

(a)

Let $f$ be a nonnegative function which is integrable over a set $E \subset \mathbb{R}^1$. Show that given $\varepsilon > 0$, there is a $\delta > 0$, such that for every set $A \subset E$ with Lebesgue measure $\vert A\vert < \delta$ we have $$\int_A f(x)\ dx < \varepsilon$$.

(b)

Let $\{f_n\}$ be a sequence of measurable functions on a set $E$. If $f_n \longrightarrow f$ in measure and there is an integrable function $g$ such that $\vert f_n\vert \le g \ \forall n$, show that $$\int_E \vert f_n-f\vert \longrightarrow 0$$. (Use (a))

四、

(a)

Let $f$ be a continuous function on $[a,b]$, and $$\int_a^b x^n f(x) \ dx = 0 \quad \forall n=1,2,3,\cdots$$. Show that $f \equiv 0$.

(b)

If $f$ is a Lebesgue integrable function on $[a,b]$ and $$\int_a^b f(x)\varphi(x)\ dx = 0$$ for all continuous function φ on $[a,b]$, what can be said about $f$, give the reasons.

五、

Let $$f(x) = \left \{ \begin{array}{ll} 0 & \mbox{for } x = 0 \\ x^{\al... ...^{-\beta}) & 0 < x \le 1 \mbox{ and } \alpha > 0, \beta > 0 \end{array} \right.$$
For what values of $\alpha, \beta$ is the function $f(x)$

(a)

continuous?

(b)

of bounded variation?

(c)

absolutely continuous?

(以上五題選做四題)

六、

(a)

Evaluate the integral $$\int_{-\infty}^{\infty} \frac{\cos x}{a^2+x^2}\ dx$$, $a$ is a positive constant.

(b)

Let $f$ be a continuous function on $[a,b]$, show that $$g(z)= \int _a^b e^{-zt} f(t) \ dt$$ is an entire function of $z$ (analytic in the entire complex plane).

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