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?



What is known about the ultra-inverse limit?
Either the even numbers or the odd numbers are $\mu$-large, so the condition degenerates to ultraproduct.


