Fatou's lemma applicability in this case?

Question

Let $\varphi: \mathbb{N} \to \mathbb{R}$; let $g: \mathbb{R}^{2} \to \mathbb{R}$; let $f_{n}: t \mapsto g(\varphi (n), t)$ on $\mathbb{R}$ for all $n \in \mathbb{N}$. Then under suitable conditions we have $$ \int \liminf_{n \to \infty}f_{n} \leq \liminf_{n \to \infty}\int f_{n} $$ by Fatou's lemma. I would like to ask if today the liminf is taken instead with respect to the argument $\varphi (n)$, what is a conclusion we can draw similar to the integral inequality? Specifically, do we still have $\int g(\liminf_{n \to \infty}\varphi (n), \cdot) \leq \liminf_{n \to \infty} \int g(\varphi_{n}, \cdot)$ to a certain extent?

Answer 1

Without continuity of $g$, we have no reason to think $g(\liminf_{n \to \infty }\varphi(n), \cdot)$ is related in any way to $g(\varphi(n), \cdot)$.