2.1.6

Posted 2020-11-09 heltion

26.设\(E\)是可测集\(,m(E)>0.\)证明存在\(x\in E,\)使得对于任意\(\delta>0,\)有\(m(E\cap B(x,\delta))>0.\)
27.若\(\{E_k\}\)是\(\mathbb R^n\)中可测集合列,证明:
(1)\(\displaystyle m(\varliminf_{k\to\infty}E_k)\leq\varliminf_{k\to\infty}m(E_k);\)
(2)若存在\(k_0,\)使得\(\displaystyle m(\bigcup_{k=k_0}^\infty E_k)<\infty,\)则\[m(\varlimsup_{k\to\infty}E_k)\geq\varlimsup_{k\to\infty}m(E_k).\]
28.设\(E\subset\mathbb R^n,H\supset E,H\)是可测集.若\(H\setminus E\)中任意可测子集皆为零测集,试问\(m(H)=m^*(E)\)吗?
29.设\(E\subset\mathbb R^n,\)证明存在\(G_\delta\)型集\(H\)且\(H\supset E,\)使得对于任意一个可测集\(A\subset\mathbb R^n,\)有\(m^*(E\cap A)=m(H\cap A).\)
30.设\(\{E_k\}\)是\([0,1]\)中可测集列\(,m(E_k)=1,k=1,2,\cdots,\)证明\[m(\bigcap_{k=1}^\infty E_k)=1.\]

26.\(\displaystyle \mathbb Q=\{r_n\},m(E)=\bigcup_{n=1}^\infty m(E\cap B(r_n,\delta)).\)
27.(1)\(\displaystyle m(\varliminf_{k\to\infty}E_k)=\lim_{k\to\infty}m(\bigcap_{j=k}^\infty E_j)\leq\lim_{k\to\infty}\inf_{j\geq k}m(E_j)=\varliminf_{k\to\infty}m(E_k).\)
(2)\(\displaystyle m(\varlimsup_{k\to\infty}E_k)=\lim_{k\to\infty}m(\bigcup_{j=k}^\infty E_j)\geq\lim_{k\to\infty}\sup_{j\geq k}m(E_j)=\varlimsup_{k\to\infty}m(E_k).\)
28.令\(G\)为\(E\)的等测包,\(m^*(E)\leq m(H)=m(H\cap G)+m(H\setminus G)\leq m(G)=m^*(E).\)
29.令\(H\)为\(E\)的等测包,\(m^*(E)=m^*(E\cap A)+m^*(E\cap A^c)=m(H)=m(H\cap A)+m(H\cap A^c).\)由\(m^*(E\cap A)\leq m(H\cap A),m^*(E\cap A^c)\leq m(H\cap A^c)\)可得\(m^*(E\cap A)=m(H\cap A).\)
30.\(\displaystyle m(\bigcap_{k=1}^\infty E_k)=m([0,1]\setminus(\bigcup_{k=1}^\infty([0,1]\setminus E_k)))=1-m(\bigcup_{k=1}^\infty([0,1]\setminus E_k))\geq1-\sum_{k=1}^\infty m([0,1]\setminus E_k)=1\)