Measure of $\partial\{g>0\}$ when $g$ is a $\mathcal C^1$ function

Question

Let $I\subset \mathbb R$ be a compact interval and $g:I\to\mathbb [0,\infty)$ a $\mathcal C^1$ function. Is it true that $$\lambda(\partial\{g>0\}) = 0,$$ where $\partial \{g>0\} := \overline{\{g>0\}}\setminus \mathrm{Int}(\{g>0\}) = \overline{\{g>0\}}\setminus \{g>0\}$ and $\lambda$ denotes the Lebesgue measure on $I$?

I think the above statement is true. However, I cannot prove it. Does anyone have a good idea?

($\lambda$ is a more common symbol for the Lebesgue measure. Still good to say what you mean by it, of course.) — aschepler, Jul 25 '23 at 13:26
That's not the definition of $\partial$. But it's ok here since ${g>0}$ is open — FShrike, Jul 25 '23 at 13:30
@FShrike I agree with you, but since $\mathrm{Int}{g>0} ={g>0}$ and I am only interested in the boundary of ${g>0}$, I decied to write like this — Matheus Manzatto, Jul 25 '23 at 13:32
A subset of $\Bbb R$ (or of $\Bbb R^n$ for that matter) is the zero set of some non-negative $C^\infty$ function if and only if it's closed. Therefore you may consider a dense open subset $\Omega$ of $I$ such that $\lambda(\Omega)\le \lambda(I)$, and $g$ such that ${g>0}=\Omega$ and ${g=0}=I\setminus\Omega$. — Sassatelli Giulio, Jul 25 '23 at 13:33
@MatheusManzatto Which result? That every closed subset is the zero set of a non-negative $C^\infty$ function, or that there are dense open sets whose complement has positive measure? — Sassatelli Giulio, Jul 25 '23 at 13:49
@SassatelliGiulio the first, that every closed subset is the zero set of a non-negative $C^∞$ function. Nevermind, this is basically smooth Uryshon's lemma https://math.stackexchange.com/questions/2891732/several-questions-on-smooth-urysohns-lemma — Matheus Manzatto, Jul 25 '23 at 13:54
See Eric Wofsey's answer to Lebesgue measure of boundary of sets of roots. Also -- Can Cantor set be the zero set of a continuous function? AND Infinitely differentiable function with given zero set? AND Every closed subset ... is the zero point set of a smooth function AND Finding a $\text C^\infty$ function which is zero just on a given closed set — Dave L. Renfro, Jul 25 '23 at 14:04

Measure of $\partial\{g>0\}$ when $g$ is a $\mathcal C^1$ function

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