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Note that even if the measure is only finitely additive, the ideal of null sets will still be countably saturated. The usual arguments involving Ulam matrices, such as can be found in Jech (Chapter 10) or other texts on set theory, show that if there is a countably additive, countably saturated ideal on some set then there is a weakly inaccessible cardinal. Hence, if no such cardinal exists then, as Emil has corrected me, the Hahn-Banach extension of Lebesgue measure will , of course, be countably additive on Lebesgue null sets but not a countably additive measureon all nulls. This does not answer your question, but does show that the weaker hypothesis still requires large cardinals.

By the way, you may want to look at the paper by Joan Hart and Ken Kunen titled "Weak Measure Extension Axioms" in Topology and Applications 85 (1998) 219-246 to see what can be said about extensions of measures to measure certain sets.

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Note that even if the measure is only finitely additive, the ideal of null sets will still be countably saturated. The usual arguments involving Ulam matrices, such as can be found in Jech (Chapter 10) or other texts on set theory, show that if there is a countably additive, countably saturated ideal on some set then there is a weakly inaccessible cardinal. Hence, if no such cardinal exists then the Hahn-Banach extension of Lebesgue measure will, of course, be countably additive on null sets but not a countably additive measure.

By the way, you may want to look at the paper by Joan Hart and Ken Kunen titled "Weak Measure Extension Axioms" in Topology and Applications 85 (1998) 219-246 to see what can be said about extensions of measures to measure certain sets.

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The usual arguments involving Ulam matrices, such as can be found in Jech (Chapter 10) or other texts on set theory, show that if there is a countably additive, countably saturated ideal on some set then there is a weakly inaccessible cardinal. Hence, if no such cardinal exists then the Hahn-Banach extension of Lebesgue measure will, of course, be countably additive on null sets but not a countably additive measure.

By the way, you may want to look at the paper by Joan Hart and Ken Kunen titled "Weak Measure Extension Axioms" in Topology and Applications 85 (1998) 219-246 to see what can be said about extensions of measures to measure certain sets.