21.證明存在開集\(G\subset\mathbb R^n,\)使\(m(\overline G)>m(G).\)
22.證明位於\(OX\)軸上任何集\(E\),在\(OXY\)平面上是可測集,且其測度為零.
23.證明有理數集是\(\mathbb R\)中可測集,且測度是\(0.\)
24.設\(E\subset\mathbb R,\)且\(m(E)>0,\)試證明存在\(x1,x_2\in E,\)使\(x_1-x_2\)是有理數.
25.證明\(E\subset\mathbb R^n\)是可測集的充分必要條件是:對於任何\(\varepsilon >0,\)存在開集\(G_1\)

21.\(\displaystyle\mathbb Q\cap[0,1]=\{r_n\},G=\bigcup_{n=1}^\infty B(r_n,\frac1{4^n}).\)
24.考慮不可測集的構造.
25.充分性\(:m(G_1\cap E^c)\leq m(G_1\cap G_2)<\varepsilon.\)
必要性\(:m(G_1\cap G_2)\leq m(G_1\cap E^c)+m(G_2\cap E)<\varepsilon.\)

2.1.5