For a (standard) $(X,A,\mu)$ be a measure space, given a function $f\colon X\to\overline{\mathbb{R}}$, we have the following characterization for its measurability. $f$ is $A$-measurable iff forall $a\in\mathbb{R}, f^{-1}([-\infty,a])\in A$.
Now, if $(Y,F,\nu)$ is a nonstandard internal measure space (i.e. $Y,F,\nu$ are internal, $F$ is an algebra and $\nu$ is only finitely additive), could we have a similar characterization for a function of kind $g:Y\to ^*\mathbb{R}$, for example $g:Y\to ^*\mathbb{R}$ is $F$-measurable iff forall $a\in\mathbb{^*R}, g^{-1}([-\infty,a])\in F$ ?
I tried with transfer principle, but the codomain of the function are different. is that proof enough?
This should follow directly from transfer, provided that $g$ is internal. The only change is that you want $g:Y\rightarrow^* \bar {\mathbb{R}}$, where $^* \bar {\mathbb{R}}$ denotes the "extended hyperreals" (i.e. you add endpoints $-^*\infty$ and $^*\infty$ to $^*\mathbb{R}$), and similarly, $g^{-1}([-^*\infty,a]\in F$.