Editing Measurable cardinal

{{Short description|Set theory concept}}
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 (mathematics)|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 [[subset]]s into large and small sets such that ''κ'' itself is large, ∅ and all [[singleton (mathematics)|singleton]]s {''α''} (with ''α''&nbsp;∈&nbsp;''κ'') are small, [[set complement|complement]]s of small sets are large and vice versa. The [[intersection]] of fewer than ''κ'' large sets is again large.<ref>{{harvnb|Maddy|1988}}</ref>

It turns out that [[uncountable]] cardinals endowed with a two-valued measure are large cardinals whose existence cannot be proved from [[ZFC]].<ref>{{harvnb|Jech|2002}}</ref>

The concept of a measurable cardinal was introduced by [[Stanisław Ulam]] in 1930.<ref>{{harvnb|Ulam|1930}}</ref>

== Definition ==
Formally, a measurable cardinal is an uncountable [[cardinal number]] ''κ'' such that there exists a ''κ''-additive, non-trivial, 0-1-valued [[measure (mathematics)|measure]] ''μ'' on the [[power set]] of&nbsp;''κ.''

Here, κ-additive means: For every ''λ''&nbsp;<&nbsp;''κ'' and every ''λ''-sized set {''A''<sub>''β''</sub>}<sub>''β''<''λ''</sub> of pairwise disjoint subsets ''A''<sub>''β''</sub>&nbsp;⊆&nbsp;''κ,'' we have
:''μ''(⋃<sub>''β''<''λ''</sub>&nbsp;''A''<sub>''β''</sub>)&nbsp;= Σ<sub>''β''<''λ''</sub>&nbsp;''μ''(''A''<sub>''β''</sub>).

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

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

== Properties ==
It is trivial to note that if ''κ'' admits a non-trivial ''κ''-additive measure, then ''κ'' must be [[regular cardinal|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''&nbsp;<&nbsp;''κ,'' then it can't be the case that ''κ''&nbsp;≤&nbsp;2<sup>''λ''</sup>. If this were the case, 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&nbsp;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 [[inaccessible cardinal|inaccessibility]].

Although it follows from [[ZFC]] that every measurable cardinal is [[inaccessible cardinal|inaccessible]] (and is [[Ineffable cardinal|ineffable]], [[Ramsey cardinal|Ramsey]], etc.), it is consistent with [[Zermelo–Fraenkel set theory|ZF]] that a measurable cardinal can be a [[successor cardinal]]. It follows from ZF&nbsp;+&nbsp;[[axiom of determinacy|AD]] that ω<sub>1</sub> is measurable,<ref name="JechDeterminacy81">T. Jech, "[https://projecteuclid.org/journalArticle/Download?urlid=bams%2F1183548432 The Brave New World of Determinacy]" (PDF download). Bulletin of the American Mathematical Society, vol. 5, number 3, November 1981 (pp.339--349).</ref> and that every subset of ω<sub>1</sub> contains or is disjoint from a [[club set|closed and unbounded]] subset.

Ulam 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.

If ''κ'' is measurable and ''p''&nbsp;∈&nbsp;''V''<sub>''κ''</sub> and ''M'' (the ultrapower of ''V'') satisfies ''ψ''(''κ,&nbsp;p''), then the set of ''α''&nbsp;<&nbsp;''κ'' such that ''V'' satisfies ''ψ''(''α,&nbsp;p'') is stationary in ''κ'' (actually a set of measure 1). In particular if ''ψ'' is a Π<sub>1</sub> formula and ''V'' satisfies ''ψ''(''κ,&nbsp;p''), then ''M'' satisfies it and thus ''V'' satisfies ''ψ''(''α,&nbsp;p'') for a stationary set of ''α''&nbsp;<&nbsp;''κ.'' This property can be used to show that ''κ'' is a limit of most types of large cardinals that are weaker than measurable. Notice that the ultrafilter or measure witnessing 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.

If one starts with an elementary embedding ''j''<sub>1</sub> of ''V'' into ''M''<sub>1</sub> with [[critical point (set theory)|critical point]] ''κ,'' then one can define an ultrafilter ''U'' on ''κ'' as {&nbsp;''S''&nbsp;⊆&nbsp;''κ''&nbsp;| ''κ''&nbsp;∈&nbsp;''j''<sub>1</sub>(''S'')&nbsp;}. Then taking an ultrapower of ''V'' over ''U'' we can get another elementary embedding ''j''<sub>2</sub> of ''V'' into ''M''<sub>2</sub>. However, it is important to remember that ''j''<sub>2</sub>&nbsp;≠&nbsp;''j''<sub>1</sub>. Thus other types of large cardinals such as [[strong cardinal]]s may also be measurable, 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 <sup>''κ''</sup>''M''&nbsp;⊆&nbsp;''M'', that is, every function from ''κ'' to ''M'' is in ''M''. Consequently, ''V''<sub>''κ''+1</sub>&nbsp;⊆&nbsp;''M''.

==Implications of existence==
If a measurable cardinal exists, every '''Σ'''{{su|b=2|p=1}} (with respect to the [[analytical hierarchy]]) set of reals has a [[Lebesgue measure]].<ref name="JechDeterminacy81" /> In particular, any [[non-measurable set]] of reals must not be '''Σ'''{{su|b=2|p=1}}.

== 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 {{harvs|txt|authorlink=Stefan Banach|first=Stefan|last=Banach|year=1930}}. {{harvtxt|Banach|Kuratowski|1929}} showed that  the [[continuum hypothesis]] implies that 𝔠 is not real-valued measurable. {{harvs|txt|authorlink=Stanislaw Ulam|first=Stanislaw|last= Ulam|year=1930}} 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 𝔠. 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 𝔠 exists if and only if there is a [[sigma additivity|countably additive]] extension of the [[Lebesgue measure]] to all sets of real numbers if and only if there is an [[atom (measure theory)|atomless]] probability measure on the power set of some non-empty set.

{{harvtxt|Solovay|1971}} showed that existence of measurable cardinals in ZFC, real-valued measurable cardinals in ZFC, and measurable cardinals in ZF, are [[equiconsistency|equiconsistent]].

=== Weak inaccessibility of real-valued measurable cardinals ===
Say that a cardinal number ''α'' is an ''Ulam number'' if<ref>{{harvnb|Federer|1996|loc=Section 2.1.6}}</ref><ref group=nb>The notion in the article [[Ulam number]] is different.</ref>

whenever
{{NumBlk|*| ''μ'' is an [[outer measure]] on a set ''X'',|{{EquationRef|1}}}}
{{NumBlk|*| ''μ''(''X'') < ∞,|{{EquationRef|2}}}}
{{NumBlk|*| ''μ''({''x''}) {{=}} 0 for every ''x'' ∈ ''X,''|{{EquationRef|3}}}}
{{NumBlk|*| all ''A '' ⊂ ''X'' are [[Carathéodory-measurable set|''μ''-measurable]],|{{EquationRef|4}}}}
then
::if |''X''| ≤ ''α'' then ''μ''(''X'') = 0.

Equivalently, a cardinal number ''α'' is an Ulam number if

whenever
# ''ν'' is an outer measure on a set ''Y,'' and ''F'' a set of pairwise disjoint subsets of ''Y,''
# ''ν''(⋃''F'') < ∞,
# ''ν''(''A'') = 0 for ''A'' ∈ ''F,''
# ⋃''G'' is ''ν''-measurable for every ''G''&nbsp;⊂&nbsp;''F,''
then
::if |''F''| ≤ ''α'' then ''ν''(⋃''F'') = 0.

The smallest infinite cardinal [[Aleph-zero|ℵ<sub>0</sub>]] is an Ulam number. The class of Ulam numbers is closed under the [[successor cardinal|cardinal successor]] operation.<ref>{{harvnb|Federer|1996|loc=Second part of theorem in section 2.1.6.}}</ref> If an infinite cardinal ''β'' has an immediate predecessor ''α'' that is an Ulam number, assume ''μ'' satisfies properties ({{EquationNote|1}})–({{EquationNote|4}}) with ''X''&nbsp;=&nbsp;''β.'' In the [[Ordinal number#Von Neumann definition of ordinals|von Neumann model]] of ordinals and cardinals, for each ''x''&nbsp;∈&nbsp;''β'', choose an [[injective function]]

:''f''<sub>''x''</sub>: ''x'' → ''α''

and define the sets

:''U''(''b,&nbsp;a'') = {&nbsp;''x''&nbsp;∈&nbsp;''β''&nbsp;| ''f''<sub>''x''</sub>(''b'')&nbsp;=&nbsp;''a''&nbsp;}

Since the ''f''<sub>''x''</sub> are one-to-one, the sets

:{&nbsp;''U''(''b, a'')&nbsp;| ''b''&nbsp;∈&nbsp;''β''&nbsp;} with ''a''&nbsp;∈&nbsp;''α'' fixed
:{&nbsp;''U''(''b, a'')&nbsp;| ''a''&nbsp;∈&nbsp;''α''&nbsp;} with ''b''&nbsp;∈&nbsp;''β'' fixed

are pairwise disjoint. By property ({{EquationNote|2}}) of ''μ,'' the set

:{&nbsp;''b''&nbsp;∈&nbsp;''β''&nbsp;| ''μ''(''U''(''b,&nbsp;a''))&nbsp;>&nbsp;0&nbsp;}

is [[countable set|countable]], and hence

:|{&nbsp;(''b, a'')&nbsp;∈&nbsp;''β''&nbsp;×&nbsp;''α''&nbsp;| ''μ''(''U''(''b, a''))&nbsp;>&nbsp;0&nbsp;}|&nbsp;≤ ℵ<sub>0</sub>⋅''α.''

Thus there is a ''b''<sub>0</sub> such that

:''μ''(''U''(''b''<sub>0</sub>, ''a'')) = 0 for every ''a''&nbsp;∈&nbsp;''α''

implying, since ''α'' is an Ulam number and using the second definition (with ''ν''&nbsp;=&nbsp;''μ'' and conditions ({{EquationNote|1}})–({{EquationNote|4}}) fulfilled),

:''μ''(⋃<sub>''a''∈''α''</sub>&nbsp;''U''(''b''<sub>0</sub>, ''a'')) = 0.

If ''b''<sub>0</sub>&nbsp;<&nbsp;''x''&nbsp;<&nbsp;''β'' and ''f''<sub>x</sub>(''b''<sub>0</sub>) = ''a''<sub>''x''</sub> then ''x''&nbsp;∈&nbsp;''U''(''b''<sub>0</sub>, ''a''<sub>''x''</sub>). Thus

:''β'' = b<sub>0</sub>&nbsp;∪ {b<sub>0</sub>}&nbsp;∪ ⋃<sub>''a''∈''α''</sub>&nbsp;''U''(''b''<sub>0</sub>, ''a'')

By property ({{EquationNote|2}}), ''μ''({''b''<sub>0</sub>})&nbsp;=&nbsp;0, and since |''b''<sub>0</sub>|&nbsp;≤&nbsp;''α'', by ({{EquationNote|4}}), ({{EquationNote|2}}) and ({{EquationNote|3}}), ''μ''(''b''<sub>0</sub>)&nbsp;=&nbsp;0. It follows that ''μ''(''β'')&nbsp;=&nbsp;0. The conclusion is that ''β'' is an Ulam number.

There is a similar proof<ref>{{harvnb|Federer|1996|loc=First part of theorem in section 2.1.6.}}</ref> that the supremum of a set ''S'' of Ulam numbers with |''S''| an Ulam number is again a Ulam number. Together with the previous result, this implies that a cardinal that is not an Ulam number is [[inaccessible cardinal|weakly inaccessible]].

== See also ==
* [[Normal measure]]
* [[Mitchell order]]
* [[List of large cardinal properties]]

== Notes ==
{{reflist|group=nb}}

== Citations ==
{{reflist}}

== References ==
*{{Citation | last1=Banach | first1=Stefan | author1-link=Stefan Banach | title=Über additive Maßfunktionen in abstrakten Mengen | url=https://eudml.org/doc/212335 | year=1930 | journal=[[Fundamenta Mathematicae]] | issn=0016-2736 | volume=15 | pages=97–101| doi=10.4064/fm-15-1-97-101 | doi-access=free }}.
*{{Citation | last1=Banach | first1=Stefan | author1-link=Stefan Banach | last2=Kuratowski | first2=Kazimierz | author2-link=Kazimierz Kuratowski | title=Sur une généralisation du probleme de la mesure | url=https://eudml.org/doc/212126 | year=1929 | journal=[[Fundamenta Mathematicae]] | issn=0016-2736 | volume=14 | pages=127–131| doi=10.4064/fm-14-1-127-131 | doi-access=free }}.
*{{Citation|author=Drake, F. R.|title=Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics; V. 76)|publisher=Elsevier Science Ltd|year=1974|isbn= 978-0-7204-2279-5}}.
*{{Citation|title=Geometric Measure Theory|series=Classics in Mathematics|edition=1st ed reprint|location=Berlin, Heidelberg, New York|orig-year=1969|year=1996|isbn=978-3540606567|publisher=[[Springer Verlag]]|last=Federer|first=H.|authorlink=Herbert Federer}}.
*{{Citation|last=Jech |first=Thomas|title=Set theory, third millennium edition (revised and expanded)|publisher=Springer|year=2002|isbn=3-540-44085-2|authorlink=Thomas Jech}}.
*{{Citation|last=Kanamori|first=Akihiro|authorlink=Akihiro Kanamori|year=2003|publisher=Springer|title=The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings|title-link=The Higher Infinite|edition=2nd|isbn=3-540-00384-3}}.
*{{Citation|last=Maddy|first=Penelope|authorlink=Penelope Maddy|journal=The Journal of Symbolic Logic|title=Believing the Axioms. II|year=1988|volume=53|issue=3|pages=736–764|doi=10.2307/2274569|jstor=2274569|s2cid=16544090}}. A copy of parts I and II of this article with corrections is available at the [http://faculty.sites.uci.edu/pjmaddy/bibliography/ author's web page].
*{{Citation | last1=Solovay | first1=Robert M. | author1-link=Robert M. Solovay | title=Axiomatic set theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) | publisher=Amer. Math. Soc. | location=Providence, R.I. |mr=0290961 | year=1971 | chapter=Real-valued measurable cardinals | pages=397–428}}.
*{{Citation | last1=Ulam | first1=Stanislaw | authorlink = Stanislaw Ulam| title=Zur Masstheorie in der allgemeinen Mengenlehre | url=https://eudml.org/doc/212487 | year=1930 | journal=[[Fundamenta Mathematicae]] | issn=0016-2736 | volume=16 | pages=140–150| doi=10.4064/fm-16-1-140-150 | doi-access=free }}.

[[Category:Large cardinals]]
[[Category:Determinacy]]
[[Category:Measures (set theory)| ]]