2016-2017第二学期实变函数期末考试

一、证明所有“有理抛物线”\(y=ax^2+bx+c\)(\(a,b,c \in \mathbb{Q}, a\neq 0\))组成的集合\(X\)是可数集.

二、设可测集\(A\subseteq [0,1]\)\(B \subseteq [1,2]\),证明\(m(A\bigcup B) = m(A) + m(B)\).

三、\(f,g\)是集合\(E\)上的可测函数,\(\forall p,q\in \mathbb{R}\),集合\(E_{pq} = \{x\in E:f(x)>p>q>g(x)\}\)为零测集,证明\(f(x) \stackrel{a.e}{\leqslant}g(x)\).

四、若在\(E\)上有\(f_n \stackrel{m}{\longrightarrow}f\)\(f_n \stackrel{a.e}{=} g_n\),证明\(g_n \stackrel{m}{\longrightarrow}f\).

五、定义在\([0,1]\)上的函数\(\displaystyle f(x)=\begin{cases}x\sin x^2,x\in \mathbb{Q}\bigcap[0,1]\\ \dfrac{x+1}{\sqrt{x}},x \in \mathbb{Q}^c\bigcap[0,1]\end{cases}\)计算积分\(\displaystyle \int_{[0,1]}f\mathrm{d}m\).

六、\(f(x)\)\((0,+\infty)\)Lebesgue可积,求\(\lim\limits_{n\rightarrow +\infty}\displaystyle \int_{(0,+\infty)}\dfrac{f(x)}{1+nx}\mathrm{d}m\).

七、\(f(x)=x^2\sin \dfrac{1}{x}\)是定义在\([0,1]\)上的函数,\(f(0)=0\),证明\(f(x)\)\([0,1]\)绝对连续.

八、定义在\([a,b]\)上的有界变差函数\(f\)满足\(\bigvee\limits_a^b(f)<1\),且\(f_n\)逐点收敛到\(f\),证明\(f_n\)也是有界变差函数.


Last update: July 26, 2020