Let $f:X \to \mathbb{R}^m$ be Borel-measurable and $z_h \in S^{m-1} \forall h \in \mathbb{N}$ (Here $S^{m-1}$ is the unit sphere in $\mathbb{R}^m$). Fix an $\epsilon>0$ and define : $\sigma : X \to \mathbb{N}$, $\sigma(x) := \min \{h\in \mathbb{N}: \langle f(x),z_h\rangle \geq (1-\epsilon)|f(x)|\} \forall x \in X$
Is $\sigma$ measurable? (The authors take this is as measurable in the text "Functions of Bounded Variation and Free Discontinuity Problems" by Luigi Ambrosio et al. page 11)
I tried to simplify the problem as follows:- $\sigma(x) := \min \{h\in \mathbb{N}: g_h(x) \geq 0\} \forall x \in X$ for some measurable functions $g_h$. Therefore, $\sigma^{-1}(h) = (g_{h_1}^{-1}[0,\infty) \cap g_{h_2}^{-1}[0,\infty) ... \cap g_{h_n}^{-1}[0,\infty)... )\cap (g_{h-1}^{-1}[0,\infty)^c \cap g_{h-2}^{-1}[0,\infty)^c ...\cap g_{0}^{-1}[0,\infty)^c) $, where $h_1, h_2, ...$ are all greater than or equal to $h$ and atleast one of them equals $h$. I see that the number of such $\{h_1,h_2,...\}$ is uncountable. So I cant prove that $\sigma$ is measurable this way. Please let me know if I can prove this any other way.
Any help is greatly appreciated.
Thanks, Phanindra