Let $\;f:\mathbb R \times \mathbb R^{n-1} \to \mathbb R^m\;$ be a continuous function and consider some bounded, non-closed and with compact closure set $\;\mathcal A\subset L^2(\mathbb R;\mathbb R^m)\;$ which consists of continuous functions $\;g:\mathbb R \to \mathbb R^m\;$.
We denote $\;d(f(\cdot,y),\mathcal A)=inf_{\;g\in \mathcal A\;} {\vert \vert f(x,y)-g(x) \vert \vert}_{L^2}\;$ and $\;B^{n-1}_R(y_0)\;$ stands for a ball in $\;\mathbb R^{n-1}\;$ of centre $\;y_0\;$ and radius $\;R\;$
Question:
if $\;d(f(\cdot,y_0),\mathcal A)\gt 0\;$ for some $\;y_0 \in \mathbb R^{n-1}\;$ then is it true to claim that $\;\mathcal L^{n-1}(B^{n-1}_R(y_0) \cap \{y: d(f(\cdot,y),\mathcal A) \ge C \gt 0\})\gt 0\;$?
My Attempt:
Since $\;\exists y_0 \in \mathbb R^{n-1}\;$ such that $\;d(f(\cdot,y_0),\mathcal A) \gt 0\;$ the set $\;B^{n-1}_R(y_0) \cap \{y: d(f(\cdot,y),\mathcal A) \ge C \gt 0\}\;$ must be non-empty.
In addition, due to continuity of $\;f\;$, there should be a neighborhood of $\;y_0\;$(let me denote it as $\;\mathcal N(y_0)\;$) in which $\;d(f(\cdot,y),\mathcal A) \ge C \gt 0\;,\forall y \in \mathcal N(y_0)\;$
Hence, if I could claim that $\;\mathcal N(y_0):=B^{n-1}_R(y_0) \cap \{y: d(f(\cdot,y),\mathcal A) \ge C \gt 0\}\;$, the question would be answered immediately taking this into account.
However as I know, a neighborhood is an open set so I'm a bit unsure if I can claim the above since I can't see whether $\;B^{n-1}_R(y_0) \cap \{y: d(f(\cdot,y),\mathcal A) \ge C \gt 0\}\;$ is a Borel one.
It's been a really long time since I had my Measure Theory course so I apologize in advance if my question is quite elementary. I would appreciate if somebody could help me understand what I'm missing here.
Thanks a lot!