Given a nonprincipal ultrafilter $\mu$ on $\mathbb{N}$ and a sequence of groups $G_i$, one can define its ultraproduct as:
$$ ^*\prod_{i\in \mathbb{N}}G_i:=\{(x_i)_{i \in \mathbb{N}}| x_i\in G_i\}/\sim$$, where $(x_i)_i\sim (y_i)_i$, iff $x_i=y_i$ $\mu$-almost everywhere.
Suppose you are given also group homomorphisms $f_{i+1}:G_{i+1}\rightarrow G_i$, then one could also consider something like an ultra- inverse limit:
$$\{[x_i]_i|f_i(x_i)=x_{i-1} \quad \mu-\mbox{almost everywhere}\}$$
Has this been studied before? Is there a good source, where I could learn about this?
