A cardinal κ is called real-valued measurable if there is an atomless κ-additive measure on the power set of κ. They were introduced by Stefan Banach (1930). Banach & Kuratowski (1929) showed that the continuum hypothesis implies that is not real-valued measurable. A real valued measurable cardinal less than or equal to exists if there is a countably additive extension of the Lebesgue measure to all sets of real numbers. A real valued measurable cardinal is weakly Mahlo.
Solovay (1971) showed that existence of measurable cardinals in ZFC, real valued measurable cardinals in ZFC, and measurable cardinals in ZF, are equiconsistent.
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