In mathematics, a measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure on a cardinal κ, or more generally on any set. For a cardinal κ, it can be described as a subdivision of all of its subsets into large and small sets such that κ itself is large, ∅ and all singletons {α} (with α ∈ κ) are small, complements of small sets are large and vice versa. The intersection of fewer than κ large sets is again large.[1]
It turns out that uncountable cardinals endowed with a two-valued measure are large cardinals whose existence cannot be proved from ZFC.[2]
The concept of a measurable cardinal was introduced by Stanisław Ulam in 1930.[3]