I have encountered the need for an unusual implicit function theorem, about which I know very little. I would appreciate it if someone could help me with a few pointers.
The setup is as follows. Let $H_i \in \mathbb{R}^{m_i\times n}$, be matrices for $i=1,...,k$. Let $M \in \mathbb{R}^{p\times n}$ be another matrix.
For each $i=1,...,k,$ let $\{g^i_n\}_{n \in \mathbb{N}}$, be a sequence of measurable functions, where $g_n^i:\mathbb{R}^{m_i} \rightarrow \mathbb{R}$. Let $f: \mathbb{R}^k \times \mathbb{R}^p \rightarrow \mathbb{R}$ be a continuous function.
Clearly, the function $y:\mathbb{R}^n \rightarrow \mathbb{R}$ defined by $y(x) := \liminf_n f(g_n^1(H_1x),...g_n^k(H_kx),Mx)$ is measurable. However, since I have taken a liminf it is not clear if $y$ can be attained by a measurable choice of the inner functions.
Specifically, I would like to know if there exist measurable functions $g^1,...,g^k$ such that $$y(x) = f(g^1(H_1x),...,g^k(H_kx),Mx) $$ for almost all $x \in \mathbb{R}^n$. It appears that I am looking for a particularly structured implicit function theorem. I believe the answer would be true if $k=1$, but I am not sure.
It would be great if someone can shed some light on this.
