Editing Measurable cardinal (section)

== 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]].