cover of $\mathbb{R^n}$

Question

Given a subset $B\subset \mathbb{R^n}$, we've said in our classes that you can write down $\mathbb{R^n}=\cup_{j\in \mathbb{N}}Q_j$ with $\lambda_n(Q_j)=1$ for all $j$. I understand the first part. But why can I write down $\lambda_n(Q_j)=1$? Is it only possible with 1 or is it possible in general with any constant $c$? ($\lambda_n$ is the Lebesgue measure of $\mathbb{R^n}$)

Answer 1

For example for $n=1$ you can consider $Q_i=[-\frac{1}{2}+\frac{i+1}{2},\frac{1}{2}+\frac{1+i}{2}]$ that has Lebesgue measure $1$ and these sets cover $\mathbb{R}$

For every $c>0$ you can consider

$Q_i=[-\frac{c}{2}+\frac{i+1}{2},\frac{c}{2}+\frac{1+i}{2}]$

Answer 2

For example let $n=2$ and for $k,m \in \mathbb Z$ let $Q_{k,m}:=[k,k+1] \times [m,m+1].$. The set $M:=\{Q_{k,m}:k,m \in \mathbb Z \}$ is countable, hence $M=\{Q_1,Q_2,...\}, \lambda_2(Q_j)=1$ and $\mathbb{R^2}=\cup_{j\in \mathbb{N}}Q_j.$