Measurable cardinal

In mathematics, a measurable cardinal is a certain kind of large cardinal number.

Measurable

Formally, a measurable cardinal is an uncountable cardinal number κ such that there exists a κ-additive, non-trivial, 0-1-valued measure on the power set of κ. (Here the term κ-additive means that, for any sequence A_α, α<λ of cardinality λ<κ, A_α being pairwise disjoint sets of ordinals less than κ, the measure of the union of the A_α equals the sum of the measures of the individual A_α.)

Equivalently, κ is measurable means that it is the critical point of a non-trivial elementary embedding of the universe V into a transitive class M. This equivalence is due to Jerome Keisler and Dana Scott, and uses the ultrapower construction from model theory. Since V is a proper class, a small technical problem that is not usually present when considering ultrapowers needs to be addressed, by what is now called Scott's trick.

Equivalently, κ is a measurable cardinal if and only if it is an uncountable cardinal with a κ-complete, non-principal ultrafilter. Again, this means that the intersection of any strictly less than κ-many sets in the ultrafilter, is also in the ultrafilter.

Although it follows from ZFC that every measurable cardinal is inaccessible (and is ineffable, Ramsey, etc.), it is consistent with ZF that a measurable cardinal can be a successor cardinal. It follows from ZF + axiom of determinacy that ω₁ is measurable, and that every subset of ω₁ contains or is disjoint from a closed and unbounded subset.

The concept of a measurable cardinal was introduced by Stanislaw Ulam (1930), who showed that the smallest cardinal κ that admits a non-trivial countably-additive two-valued measure must in fact admit a κ-additive measure. (If there were some collection of fewer than κ measure-0 subsets whose union was κ, then the induced measure on this collection would be a counterexample to the minimality of κ.) From there, one can prove (with the Axiom of Choice) that the least such cardinal must be inaccessible.

It is trivial to note that if κ admits a non-trivial κ-additive measure, then κ must be regular. (By non-triviality and κ-additivity, any subset of cardinality less than κ must have measure 0, and then by κ-additivity again, this means that the entire set must not be a union of fewer than κ sets of cardinality less than κ.) Finally, if λ < κ, then it can't be the case that κ ≤ 2^λ. If this were the case, then we could identify κ with some collection of 0-1 sequences of length λ. For each position in the sequence, either the subset of sequences with 1 in that position or the subset with 0 in that position would have to have measure 1. The intersection of these λ-many measure 1 subsets would thus also have to have measure 1, but it would contain exactly one sequence, which would contradict the non-triviality of the measure. Thus, assuming the Axiom of Choice, we can infer that κ is a strong limit cardinal, which completes the proof of its inaccessibility.

If κ is measurable and p∈V_κ and M (the ultrapower of V) satisfies ψ(κ,p), then the set of α<κ such that V satisfies ψ(α,p) is stationary in κ (actually a set of measure 1). In particular if ψ is a Π₁ formula and V satisfies ψ(κ,p), then M satisfies it and thus V satisfies ψ(α,p) for a stationary set of α<κ. This property can be used to show that κ is a limit of most types of large cardinals which are weaker than measurable. Notice that the ultrafilter or measure which witnesses that κ is measurable cannot be in M since the smallest such measurable cardinal would have to have another such below it which is impossible.

Every measurable cardinal κ is a 0-huge cardinal because ^κM⊂M, that is, every function from κ to M is in M. Consequently, V_κ+1⊂M.

Real-valued measurable

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 ${\mathfrak c}$ is not real-valued measurable. A real valued measurable cardinal less than or equal to ${\mathfrak c}$ 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.

See also

• Normal measure
• Mitchell order

References

• Banach, Stefan (1930), "Über additive Massfunktionen in absrakten Mengen", Fundamenta Mathematicae 15: 97–101, ISSN 0016-2736, http://matwbn.icm.edu.pl/tresc.php?wyd=1&tom=15 
• Banach, Stefan; Kuratowski, C. (1929), "Sur une généralisation du probleme de la mesure", Fundamenta Mathematicae 14: 127–131, ISSN 0016-2736, http://matwbn.icm.edu.pl/tresc.php?wyd=1&tom=14 
• Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics ; V. 76). Elsevier Science Ltd. ISBN 978-0720422795. 

• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed ed.). Springer. ISBN 3-540-00384-3. 
• Solovay, Robert M. (1971), "Real-valued measurable cardinals", Axiomatic set theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967), Providence, R.I.: Amer. Math. Soc., pp. 397–428, MR0290961 
• Ulam, Stanislaw (1930), "Zur Masstheorie in der allgemeinen Mengenlehre", Fundamenta Mathematicae 16: 140–150, ISSN 0016-2736, http://matwbn.icm.edu.pl/tresc.php?wyd=1&tom=16