Originally Posted by

**Opalg** 1) Given m, n and k, the box $\displaystyle [m,m+1]\times[n,n+1]\times[-1/k,1/k]$ has volume 2/k. Taking the intersection over all natural numbers k, you see that the set $\displaystyle \{(x,y,z)\in\mathbb{R}^3:m\leqslant x\leqslant m+1,\ n\leqslant y\leqslant n+1,\ z=0\}$ has measure 0. Now take the (countable) union over all integers m and n.

2) By the same argument as in 1), the set $\displaystyle \{(x,y,z,t)\in\mathbb{R}^4:z=r\}$ has measure 0, for each rational number r. Now take the union over all such r.

Opalg's eplanation is clear. If I understand Opalg's 1), then by parrallel argument, should 2) be unioned over three indexes?:

2) By the same argument as in 1), the set $\displaystyle \{(x,y,z,t)\in\mathbb{R}^4:t=r\}$ has measure 0, for any rational number r. Now take the (countable) union over all integers m, n and p.