Outer measure and null sets

Question

This seems to hold true for lebeesgue outer measure, or is my proof wrong?:

If $m^*(N)=0$, then let $F \subseteq \mathbb{R^d}$, and we have $m^*(F\setminus N) = m^*(F)$.

Proof: Let us use lebesgue outer measure definition on real line extended similarly to $\mathbb{R}^d$. Clearly, we have inequality one one side, and by subadditivity we have $$m^*(F) \le m^*(F\setminus N) + m^*(N) = m^*(F\setminus N).$$

Answer 1

Of course, the second inequality is trivial by monotonicity of outer measure. Your argument works for any outer measure, not necessarily the Lebesgue one.