Outer measure of $\mathbb R^*$

Question

I know outer measure of $\mathbb R$ is $\infty$ But what about outer measure of $\mathbb R^*$ is also $\infty$ ?

My answer:

$\mathbb R^*=\mathbb R$\ {$0$}$ =(-\infty,0) \cup (0,\infty)$

$m^*(\mathbb R^*) \leq m^*((-\infty,0) \cup (0,\infty))$

$= \infty + \infty = \infty$

True ?