In Section 13 of Probability and Measure by Billingsley, it has been shown that for a measurable space $(F, \mathcal{F})$, $g:F\rightarrow \mathbb{R}^m$ and $g_i: F\rightarrow \mathbb{R}$ with $g(x) = [g_1(x),\cdots, g_m(x)], \forall x \in F$, $g$ is $\mathcal{F}/\mathcal{B}(\mathbb{R}^m)$ measurable if and only if $g_i$ is $\mathcal{F}/\mathcal{B}(\mathbb{R})$ measurable.
More generally, suppose $\{ (G_i, \mathcal{G}_i), i=1,\cdots,m \}$ are measurable space, $(\prod_{i=1}^m G_i, \mathcal{G})$ is also a measurable spaces, but $\mathcal{G}$ may not be $\prod_{i=1}^m \mathcal{G}_i$. for $g:F \rightarrow \prod_{i=1}^m G_i$, and $g_i: F \rightarrow G_i$ with $g(x) = [g_1(x),\cdots, g_m(x)]$. I wonder what are some conditions under which $g$ is measurable if and only if $g_i: F \rightarrow G_i, i=1, \cdots, m$ are measurable? Can the fact $g(x)=[g_1(x),⋯,g_m(x)]$ be used in your conditions?
How will your conditions be used to explain the example when $(G_i, \mathcal{G_i}) = (\mathbb{R}, \mathcal{B}(R))$ and $(\prod_{i=1}^m G_i, \mathcal{G}) = (\mathbb{R}^m, \mathcal{B}(\mathbb{R}^m))$ mentioned earlier? Thanks and regards!