computable $sigma$-aglebra and countably generated $sigma$-algebra

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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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    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)




    measure-theory




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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)




      measure-theory




      share|cite|improve this question











      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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      asked Jul 20 at 5:20









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