臺灣大學數學系

八十九學年度第一學期碩博士班資格考試試題

 分析

[回上頁]

一、

Let $E$ be a measurable set. $\int_A f = 0$ for all measurable subsets $A$ of $E$. Show that $f=0 \quad a.e.$ $(E)$.

二、

$E \subset R$ is measurable. $f$ is integrable on $E$. (Lebesgue)

(a)

Then so is $\vert f\vert$ and $\Big \vert \int_E f \Big \vert \le \int_E \vert f\vert$?

(b)

Consider the improper Riemann integral of $f$ (of a fixed kind) on $[a,b]\quad (a < b \le \infty)$ write $$\mbox{\bf Imp} f = \int_a^b f$$. (Riemann)
If $f$ is Lebesgue integrable on $[a,b]$, then what conclusions can you make about $I_m f$?

三、

Let $g(t)$ be monotonic increasing and absolutely continuous on $[\alpha, \beta]$. $f$ bounded measurable on $[a,b]$ with $a=g(\alpha), b = g(\beta)$. Show that $f(g(t))$ is measurable on $[\alpha, \beta]$ and

$$\int_a^b f(x) \ dx = \int_{\alpha}^{\beta} f(g(t)) g'(t)\ dt$$

四、

Suppose that $f_k \longrightarrow f \quad a.e.$ in $R$ and $f_k, f \in L^p(R) \quad 1 < p < \infty$. If $\Vert f_k \Vert _p \le M < \infty$, show that $\int f_k g \longrightarrow \int fg \quad \forall g \in L^q, \frac1p+\frac1q = 1$.

五、

Assume $f \in L^1 (R)$ then $$\lim\limits_{n \rightarrow \infty} \sum_{k = -n^2}^{n^2} \Big \vert \int_{\frac{k}{n}}^{\frac{k+1}{n}} f(x)\ dx \Big \vert=$$? Give reasons.

六、

(a)

Describle Cauchy's integral formula.

(b)

Show that bounded entire function (on $C$) reduces to constant.

七、

Calculate $$\int_0^{\infty} \frac{\cos mx}{1+x^2} \ dx$$, where $m > 0$ is a constant.

[回上頁]