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Oct 28, 2016 at 3:26 comment added Nate Eldredge I also don't know anything about random topologies, but there are notions of random metric spaces. Google can tell you as much as I know.
Oct 28, 2016 at 3:24 comment added Nate Eldredge Well, I can't say I've seen anything like this before. But if it were me, rather than viewing it as an operation-valued random variable, I'd focus instead on the map $F : X \times X \times I \to X$ which sends $(x,y,i)$ to what you call $x \otimes^i y$. Then all you need are $I$-valued random variables, or equivalently, probability measures on $I$ or $I^\omega$.
Oct 27, 2016 at 18:48 history edited Jeyrome Sapin CC BY-SA 3.0
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Oct 27, 2016 at 18:21 history edited Jeyrome Sapin CC BY-SA 3.0
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Oct 27, 2016 at 18:17 comment added Jeyrome Sapin @NateEldredge : I edited my post.
Oct 27, 2016 at 18:15 history edited Jeyrome Sapin CC BY-SA 3.0
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Oct 27, 2016 at 17:01 comment added Nate Eldredge Do you have in mind a concrete example where such a construction would be interesting?
Oct 27, 2016 at 15:46 history edited Jeyrome Sapin CC BY-SA 3.0
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Oct 27, 2016 at 15:18 history edited Jeyrome Sapin CC BY-SA 3.0
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Oct 27, 2016 at 13:10 history edited Jeyrome Sapin CC BY-SA 3.0
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Oct 27, 2016 at 12:47 history asked Jeyrome Sapin CC BY-SA 3.0