Let $X$ be a random vector in $\mathbb{R}^{d}$, and let $f_{X}:\mathbb{R}^{d}\rightarrow \mathbb{R}_{+}$ and $g_{X}:\mathbb{R}^{d}\rightarrow \mathbb{R}_{+}$ two functions such that $f_{X}$ and $g_{X}$ depend on $X$.
Suppose that, for all $\epsilon>0$ and $0<\delta<1$, we have $$\mathbb{P}\left(|f_{X}(x)-g_{X}(x)|<\epsilon \right)\geq 1-\delta \quad \forall x\in \mathbb{R}^{d}. \tag{1}$$
What can I say about the relationship between $\underset{x}{\mathrm{argmax}}f_{X}(x)$ and $\underset{x}{\mathrm{argmax}}g_{X}(x)$?
Remark: From my point of view, as a product of my intuition, and taking into account the fact (1) for very small values of $\epsilon>0$, what should happen is that, given any $\kappa>0$, for all $x^{*}\in \underset{x}{\mathrm{argmax}}f_{X}(x)$ there should exist $x^{'}\in \underset{x}{\mathrm{argmax}}g_{X}(x)$ such that $\|x^{*}-x^{'}\|<\kappa$, all this with probability at least $1-\delta$. In probabilistic terms, this translates to $$ \mathbb{P}\left(\bigcup_{x^{*}\in \underset{x}{\mathrm{argmax}}f_{X}(x)}\bigcap_{x^{'}\in \underset{x}{\mathrm{argmax}}g_{X}(x)}\|x^{*}-x^{'}\|<\kappa\right)\geq 1-\delta $$ However, you will note that in this probability there may appear unions and intersections indexed on non-countable sets, so this probability seems not to be easy to calculate.
I would like to know what you think about this problem and your proposals.