Show that the $2$-dimensional outer measure $m^\ast_2(B) =0$, when $B=\{(x,x) \mid 0 \le x \le 1 \}$.

Question

Show that the $2$-dimensional outer measure $m^\ast_2(B) =0$, when $B=\{(x,x) \mid 0 \le x \le 1 \}$.

Here $m^\ast_2(B) = \{ \sum_{I \in \mathcal{F}} \ell(I) \mid \text{$\mathcal{F}$ covers $B$}\}$

Set $B$ seems to form a diagonal line from $0$ to $1$. Now It seems I should construct an cover $\mathcal{F}$ for $B$ in such a way that $\sum_{I \in \mathcal{F}} \ell(I) \le \varepsilon$.

Any hints on how to approach the construction of $\mathcal{F}$ here?