Let $w \in C^1(\mathbb R^n)$ be given, such that the set $M := \{x \in \mathbb R^n \mid w(x) = 0\}$ is compact. Moreover, we assume that $\nabla w(x) \ne 0$ for all $x \in M$. This means that $M$ is a compact, differentiable, $(n-1)$-dimensional submanifold.
Now, we consider $z \in C^1(\mathbb R^n)$ and the perturbed set $M_z := \{x \in \mathbb R^n| w(x) + z(x) = 0\}$. If $z$ and $\nabla z$ are small enough on $\mathbb R^n$, we again have $\nabla w(x) + \nabla z(x) \ne 0$ for all $x \in M_z$, i.e., $M_z$ is again a manifold.
Intuitively, we have that $M_z$ converges towards $M$. More specifically, I am interested in the convergence of integrals, i.e., for a given $g \in C(\mathbb R^n)$, do we have $$\int_{M_z} g \, \mathrm d \mathcal H^{n-1} \to \int_M g \, \mathrm d \mathcal H^{n-1}$$ as $z \to 0$ (in a suitable sense, e.g., $z, \nabla z \to 0$ uniformly on $\mathbb R^n$)?
Here, $\mathcal H^{n-1}$ is the $(n-1)$-dimensional Hausdorff measure (or, equivalently, the surface measure of the manifolds).
I am pretty sure that this result should be somewhere in the literature. References are welcome!