Let $\mu$ be a probability measure over the (closed but unbounded) set $X \subseteq \mathbb{R}^m$: $\int_X \mu(dx) = 1$.
Consider function $f:\mathbb{R}^n \times \mathbb{R}^n \times X \rightarrow \mathbb{R}_{\geq 0}$ that is:
- continuous in the first argument;
- locally bounded in the second argument.
Moreover, function $x \mapsto f(y,y,x) $ is integrable for all $y \in \mathbb{R}^n$.
Consider a sequence $\{z_i\}_{i=1}^{\infty}$ such that: $z_j \in \mathbb{R}^n$, $z_j \rightarrow z$.
I would like to state the following result.
$ \forall \epsilon \in \mathbb{R}_{>0} \quad \exists i^* \in \mathbb{N} $ such that:
$ \displaystyle \int_{X} | f(z,z_i,x) - f(z_i,z_i,x) | \mu(dx) \ \leq \ \epsilon \qquad \forall i \geq i^*. $
Proof?
Additional notes: if the result is not true, I'm interested in both finding a counterexample and finding additional assumptions that would lead the result.