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• W3148730244 abstract "We investigate which infinite binary sequences (reals) are effectively random with respect to some continuous (i.e., non-atomic) probability measure. We prove that for every <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, all but countably many reals are <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-random for such a measure, where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> indicates the arithmetical complexity of the Martin-Löf tests allowed. The proof rests upon an application of Borel determinacy. Therefore, the proof presupposes the existence of infinitely many iterates of the power set of the natural numbers. In the second part of the paper we present a metamathematical analysis showing that this assumption is indeed necessary. More precisely, there exists a computable function <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=application/x-tex>G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that, for any <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the statement “All but countably many reals are <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G left-parenthesis n right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>G(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-random with respect to a continuous probability measure” cannot be proved in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sans-serif upper Z sans-serif upper F sans-serif upper C Subscript n Superscript minus> <mml:semantics> <mml:msubsup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=sans-serif>Z</mml:mi> <mml:mi mathvariant=sans-serif>F</mml:mi> <mml:mi mathvariant=sans-serif>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> </mml:msubsup> <mml:annotation encoding=application/x-tex>mathsf {ZFC}^-_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Here <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sans-serif upper Z sans-serif upper F sans-serif upper C Subscript n Superscript minus> <mml:semantics> <mml:msubsup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=sans-serif>Z</mml:mi> <mml:mi mathvariant=sans-serif>F</mml:mi> <mml:mi mathvariant=sans-serif>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> </mml:msubsup> <mml:annotation encoding=application/x-tex>mathsf {ZFC}^-_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> stands for Zermelo-Fraenkel set theory with the Axiom of Choice, where the Power Set Axiom is replaced by the existence of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-many iterates of the power set of the natural numbers. The proof of the latter fact rests on a very general obstruction to randomness, namely the presence of an internal definability structure." @default.
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• W3148730244 date "2021-08-30" @default.
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• W3148730244 title "Effective randomness for continuous measures" @default.
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• W3148730244 doi "https://doi.org/10.1090/jams/980" @default.
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