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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?

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I've never seen this before, but it looks like a fascinating idea. Are you taking this as a subgroup of the Cartesian product, or a subgroup of the ultraproduct? One guess I would make is that it won't be possible to put a nice topology on this construction, in the way that e.g. profinite groups have a nice topology. –  Colin Reid Oct 21 '10 at 12:39
I'll bet he means a subgroup of the ultraproduct; the square brackets in the definition surely indicate an equivalence class? –  Harald Hanche-Olsen Oct 21 '10 at 13:41
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1 Answer

up vote 6 down vote accepted

Either the even numbers or the odd numbers are $\mu$-large, so the condition degenerates to ultraproduct.

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OK this explains why I didn't find anything about it –  HenrikRüping Oct 21 '10 at 13:43
But does the question still make sense for more general ultra-inverse limits (on large index sets, etc.)? –  Andrés Villaveces Mar 16 '12 at 4:37
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