Lebesgue measure simple property. Seems intuitevely true but can't formalise a proof.

Consider the usual Lebesgue measure in $\mathbb R^n$ and let $f,g \colon \mathbb R^n \to \mathbb R$ be two arbitrary functions that satisfy

$$ | \{ x \in \mathbb R^n : g(x) \neq h(x) \} | > 0 \quad \text{ and }\quad | \{x \in \mathbb R^n : h(x) \neq 0\}| = 0, $$

where $|A|$ stands for the Lebesgue measure of the subset $A \subset \mathbb R^n$.

GOAL. My aim is to prove that, under these hypothesis, there holds

$$ | \{ x \in \mathbb R^n : g(x) \neq 0\}| > 0. $$

My attempt. Intuitively, this implication seems true but I am having a hard time proving it formally. I appreciate any hints so that I can begin my reasoning.

Thanks for any help in advance.