measurable and real-valued measurable cardinals

Let κ be an uncountable cardinal. Then

  1. 1.

    κ is measurable if there exists a nonprincipal κ-completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ultrafilter U on κ;

  2. 2.

    κ is real-valued measurable if there exists a nontrivial κ-additive measure μ on κ.

If κ is measurable, then it is real-valued measurable. This is so because the ultrafilter U and its dual ideal I induce a two-valued measure μ on κ where every member of U is mapped to 1 and every member of I is mapped to 0. Since U is κ-complete, I is also κ-complete. It can then be proved that if Iμ–the ideal of those sets whose measures are 0–is κ-complete, then Iμ is κ-additive.

On the converseMathworldPlanetmath side, if κ is not real-valued measurable, then κ20. It can be shown that if κ is real-valued measurable, then it is regularPlanetmathPlanetmath; a further result is that κ is weakly inaccessible. Inaccessible cardinals are in some sense ”large.”

Title measurable and real-valued measurable cardinals
Canonical name MeasurableAndRealvaluedMeasurableCardinals
Date of creation 2013-03-22 18:54:53
Last modified on 2013-03-22 18:54:53
Owner yesitis (13730)
Last modified by yesitis (13730)
Numerical id 4
Author yesitis (13730)
Entry type Definition
Classification msc 03E55