What follows will be abstract nonsense: Let $d\in\mathbb N$ and $\iota_d:\mathbb R\to\mathbb R^d$ such that ${\iota_d(x)}_1=x$ for all $x\in\mathbb R$. $\iota$ is an embedding (i.e. an injective function) of $\mathbb R$ into $\mathbb R^d$.
Now, let $\pi_d:\mathbb R^d\to\mathbb R$ denote the projection onto the first component. If $(\Omega,\mathcal A)$ is a measurable space and $f_d:\Omega\to\mathbb R^d$ is $(\mathcal A,\mathcal B(\mathbb R^d))$-measurable, we obtain a $(\mathcal A,\mathcal B(\mathbb R)$-measurable function via $$g_d:=\pi_d(f_d).$$
My question is: Are we able to generalize the considerations above to a sequence of metric spaces $E_d$ such that $E_1$ is embedded into $E_d$ via $\iota_d$ for $d\in\mathbb N$?
Intuitively, there should be some analogue of $\pi_d$ in that case too. What I want is that if $f_d:\Omega\to E_d$ is $(\mathcal A,\mathcal B(E_d))$-measurable, $g_d=\pi_d(f_d)$ is $(\mathcal A,\mathcal B(E))$-measurable.
Obviously, in the example of $\mathbb R^d$ at the beginning of the question, $\iota_d$ and $\pi_d$ are somehow related (while $\iota_d$ is clearly not unique).
Feel free to add assumptions on $E_d$ and $\iota_d$.
(Clearly, everything generalizes literally if $E_d=E^d$ for some Polish space $E$; since then $\mathcal B(E_d)={\mathcal B(E)}^{\otimes d}$.)