Measurable cardinal
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In mathematics, a measurable cardinal is a certain kind of large cardinal number.
Formally, a measurable cardinal is a cardinal number κ such that there exists a κ-additive, non-trivial, 0-1-valued measure on the power set of κ. Equivalently, κ is measurable if it is the critical point of a non-trivial elementary embedding of the universe V into a transitive class M.
A cardinal κ is real-valued measurable iff there is an atomless κ-additive measure on the power set of κ. A real valued measurable cardinal is weakly Mahlo and has cardinality less than or equal to that of the continuum. Thus, existence of real valued measurable cardinals would imply the negation of the continuum hypothesis. A real valued measurable cardinal exists iff there is a countably additive extension of the Lebesgue measure to all sets of real numbers.
A regular cardinal is measurable iff there is a cardinal preserving generic extension of V in which the cardinal is singular.
Although in ZFC every measurable cardinal is inaccessible (and is ineffable, Ramsey, etc.), a measurable cardinal in ZF can be a successor. Under ZF + axiom of determinacy, ω1 is measurable, and every subset of ω1 contains or is disjoint from a closed and unbounded subset.
Existence of measurable cardinals in ZFC, real valued measurable cardinals in ZFC, and measurable cardinals in ZF is equiconsistent.