but not using the same embedding. It can be shown that a strong cardinal κ is measurable and also has κ-many measurable cardinals below it. Every measurable cardinal κ is a 0-huge cardinal because κM⊆M, that is, every function from κ to M is in M. Consequently, Vκ+1⊆M. Implications of existence If a measurable cardinal exists, every (with respect to the Borel hierarchy) set of reals has a Lebesgue measure. In particular, any non-measurable set of reals must not be . Real-valued measurable A cardinal κ is called real-valued measurable if there is a κ-additive probability measure on the power set of κ that vanishes on singletons. Real-valued measurable cardinals were introduced by . showed that the continuum hypothesis implies that is not real-valued measurable. showed (see below for parts of Ulam's proof) that real valued measurable cardinals are weakly inaccessible (they are in fact weakly Mahlo). All measurable cardinals are real-valued measurable, and a real-valued measurable cardinal κ is measurable if and only if κ is greater than . Thus a cardinal is measurable if and only if it is real-valued measurable and strongly inaccessible. A real valued measurable cardinal less than or equal to exists if and only if there is a countably additive extension of the Lebesgue measure to all sets of real numbers if and only if there is an atomless probability measure on the power set of some non-empty set. showed that existence of measurable cardinals in ZFC, real valued measurable cardinals in ZFC, and measurable cardinals in ZF, are equiconsistent. Weak inaccessibility of real-valued measurable cardinals Say that a cardinal number is an Ulam number if whenever then Equivalently, a cardinal number is an Ulam number if whenever is an outer measure on a set , and a disjoint family of subsets of , for is -measurable for every then The smallest infinite cardinal is an Ulam number. The class of Ulam numbers is closed under the cardinal successor operation. If an infinite cardinal has an immediate predecessor that is an Ulam number, assume satisfies properties ()–() with . In the von Neumann model of ordinals and cardinals, choose injective functions and define the sets Since the are one-to-one, the sets are disjoint. By property () of , the set is countable, and hence Thus there is a such that implying, since is an Ulam number and using the second definition (with and conditions