Effective randomness for continuous measures

We investigate which infinite binary sequences (reals) are effectively random with respect to some continuous (i.e., non-atomic) probability measure. prove that for every n n , all but countably many reals -random such a measure, where indicates the arithmetical complexity of Martin-Löf tests allowed. The proof rests upon an application Borel determinacy. Therefore, presupposes existence infinitely iterates power set natural numbers. In second part paper we present metamathematical analysis showing this assumption is indeed necessary. More precisely, there exists computable function alttext="upper G"> G encoding="application/x-tex">G that, any statement “All G left-parenthesis n right-parenthesis"> ( stretchy="false">) encoding="application/x-tex">G(n) measure” cannot be proved in alttext="sans-serif upper Z sans-serif F C Subscript Superscript minus"> Z mathvariant="sans-serif">F mathvariant="sans-serif">C − encoding="application/x-tex">\mathsf {ZFC}^-_n . Here stands Zermelo-Fraenkel theory Axiom Choice, Power Set replaced by -many latter fact on very general obstruction randomness, namely presence internal definability structure.