2.1.5

Posted 2020-11-08 heltion

21.证明存在开集\(G\subset\mathbb R^n,\)使\(m(\overline G)>m(G).\)
22.证明位于\(OX\)轴上任何集\(E\),在\(OXY\)平面上是可测集,且其测度为零.
23.证明有理数集是\(\mathbb R\)中可测集,且测度是\(0.\)
24.设\(E\subset\mathbb R,\)且\(m(E)>0,\)试证明存在\(x1,x_2\in E,\)使\(x_1-x_2\)是有理数.
25.证明\(E\subset\mathbb R^n\)是可测集的充分必要条件是:对于任何\(\varepsilon >0,\)存在开集\(G_1\)与\(G_2,G_1\supset E,G_2\supset E^c,\)使得\[m(G_1\cap G_2)<\varepsilon.\]

21.\(\displaystyle\mathbb Q\cap[0,1]=\{r_n\},G=\bigcup_{n=1}^\infty B(r_n,\frac1{4^n}).\)
24.考虑不可测集的构造.
25.充分性\(:m(G_1\cap E^c)\leq m(G_1\cap G_2)<\varepsilon.\)
必要性\(:m(G_1\cap G_2)\leq m(G_1\cap E^c)+m(G_2\cap E)<\varepsilon.\)