I needed help in showing that the set $R^{n-1} \times 0$ has measure zero in $R^n$.
What I have so far: Let $\epsilon > 0$. If $i_1,\dots,i_{n-1}$ are integers, then define $U_{i_1,\dots,i_{n-1}}=[i_1,i_1+1]\times \cdots \times [i_{n-1},i_{n-1}+1]$. Chose a bijection $f:Z\times Z\times \cdots \times Z\to N$ (the product has $n-1$ factors) where $N$ is the set of positive integers and define $A_{i_1,\dots,i_{n-1}}=U_{i_1,\dots,i_{n-1}} \times [-2^{-f(i_1,\dots,i_{n-1})-1}\epsilon,2^{-f(i_1,\dots,i_{n-1})-1}\epsilon]$. The collection of all such $A_{i_1,\dots,i_{n-1}}$'s covers $R^{n-1}\times 0$ and the sum of the measures of the $A_{i_1,\dots,i_{n-1}}$'s is less than $\epsilon$. (Why?)