Let $x \in D$ where $D$ is a compact domain. Consider a smooth function $f(x)$ and another function $f_n(x)$ (a sequence of functions indexed by $n$) such that $f_n \rightarrow f$ uniformly as $n \rightarrow \infty$. I am interested in bounding the following difference: $\int\mathbb{1}_{\{x:f(x) > \epsilon\}} \, g(x) \, dx - \int \mathbb{1}_{\{x: f_n(x) > \epsilon\}} \,g(x) \,dx$
for some smooth function $g(x)$ and $\epsilon$. Suppose that $\{x: f(x) > \epsilon\}$ and $\{x: f_n(x) > \epsilon\}$ are non-empty for sufficiently large $n$. Is there any technique I can invoke to replace the difference of indicator functions into analytic expressions that are in a sense related to $f(x)$ and $f_n(x)$ which are more tractable to work with? You can say that I am looking to show that as $n \rightarrow \infty$, both quantities are equal but I'd like for the upper bound to depend explicitly how $f_n$ approximates $f$.
I have seen a technique involving expressing the expression above using surface integrals but I don't know what the technique is called nor if I can apply it to my case as it is a slightly different expression. In it, quantity they bound looks like:
$\int \mathbb{1}_{\{x: T(x) < \epsilon + \mu\}} \, g(x) \, dx - \int \mathbb{1}_{\{x: T(x) < \epsilon - \mu\}} \, g(x) \, dx$ which turns out to be equal to $\int \mathbb{1}_{\{x: T(x) \in (\epsilon-\mu,\epsilon+\mu)\}} \, g(x) \, dx$ after which it was written as a surface integral of the form:
$\int \mathbb{1}_{T(x^*) = \epsilon} (h^1(x^*) - h^2(x^*))\, g(x^*) \,dS$ where $h^1$ and $h^2$ satisfy an equation related to the surface $T(x^*) = \epsilon$.
Are there any similar tools available to me?