台灣大學數學系

九十一學年度第二學期博士班資格考試題

實分析

 

May 10, 2003

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There are 12 problems,do any 10 of them.

Let $(X,\mathbf{B},\mu)$ be a $\sigma$-finite measure space (i.e. there exist countably many $X_j$ $\in$ $\mathbf{B}$
such that $\mu X_j<\infty$ and $X=\cup_j X_j$).Let $f \in L^1$ $(\mu)$
(1)Prove for any $\epsilon>0$, there is a measurable function $g$ vanishing outside a set of finite
measure such that $\int\vert f-g\vert d\mu<\epsilon$.
(2)Prove for any $\epsilon>0,\exists\delta>0$ such that $\int_A\vert f\vert d\mu<\epsilon$ for any A $\in \mathbf{B}$ with $\mu A<\delta$.
(3)Define a signed measure $\nu$ by $\nu$ E= $\int_E f d \mu$. Let$\{E_n\}$ be a decreasing sequence of
measurable sets. Prove $\nu(\cap^\infty_1E_j)=lim_{j\rightarrow\infty}\nu E_j$ .
(4)If $X=[0,1]$ and $\mu$ is the Lebesgue measure , prove $F(x)$ defined by $F(x)=\int^x_0 f(t)dt$
is absolutly continuous.
Let $f$ be a real-valued measurable function on [a,b] and $E=\{x:\vert f'(x)\vert< \alpha\}$.Let
$m^*$ and $m$ be respectively the Lebesgue outer measure and Lebesgue measure on [a,b].
(5) If $f$ is absolutely continuous, prove $m^* (f(E))\leq \alpha m^*(E)$.
(6) If $f$ is of bounded variation , is it still true that $m^* (f(E))\leq \alpha m^*(E)$ ?Prove or
disprove your answer .

Let $X$ be a compact subset of $\mathbf{R}^n$ . Let $f_n \in L^p(X) (p>1)$ be a sequence converging
a.e. to a measurable function $f$. Suppose $\Vert f_n-f\Vert _p < M$ for some $M$ .
(7) Prove $f \in L^p(X)$ and $f_n$ converges to $f$ weakly .
(8) Prove $f_n$ converges to $f$ in $L^r(X)$ for any $1\leq r \leq p$ .

Let $\mu$ be a measure on the compact subset $K$ of $\mathbf{C}$ . Define $f(z)=\int_K \frac{d \mu (\zeta)}{\zeta-z}$ .
(9) Prove $f$ is analytic outside $K$ .
(10) Find an estimate for $\vert f'(z)\vert$ for $z$ not in $K$ .

Let $\phi$ be a bounded measurable function defined on $\{(x,y) : \vert x\vert^2+\vert y\vert^2 \leq 1\}\subset \mathbf{R}^2$ .
Define for $(x,y) \in \mathbf{R}^2$

$$g(x,y)= \int_ {\vert s\vert^2+\vert t\vert^2 \leq 1} \phi (s,t)\log ((s-x)^2+(t-y)^2)dm,$$

where $m$ is the Lebesgue measure on $\mathbf{R}^2$ .
(11) Prove $g$ is locally integrable hence finite a.e..
(12) Prove the partial derivatives of $g$ exist .

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