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It seems that it is built into the definition of a computable measure space that the $\sigma$-algebra is countably generated (by a countable ring). I am wondering if there is a general notion of computability that allows us to talk about computable $\sigma$-algebra in abstract, and whether there are uncountably generated computable $\sigma$-algebra (I gathered from here (Are there any uncountable sets that are computable?) that computability could make sense when applied to uncountable sets)

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