Question.Is it consistent with ZF thatevery(countably additive, non-negative) measure $\mu: \Sigma \to \bf R$, where $\Sigma$ is a sigma-algebra on a given set $X$, extends to a (countably additive, non-negative) measure $\tilde{\mu}: \mathcal P(X) \to \bf R$?

What I seem to know: Let us first work in the frame of ZFC. Given a set $X$ of cardinality $\ge \aleph_1$, we take $\Sigma$ to be the smallest sigma-algebra on $X$ containing all the countable subsets of $X$. It is seen that the function $$\mu: \Sigma \to {\bf R}: A \mapsto \left\{ \begin{array}{ll} \! 0 & \text{if } |A| \le \aleph_0 \\ \! 1 & \text{otherwise} \end{array} \right. $$ is a measure on $X$. Thus, if $\mu$ extends to a (countably additive, non-negative) measure $\mathcal P(X) \to \bf R$, then $|X|$ is a real-valued measurable cardinal, which implies the existence of a weakly inaccessible cardinal by an old theorem of S. Ulam, see

S. Ulam,

Zur Masstheorie in der allgemeinen Mengenlehre, Fund. Math.16(1930), 140-150 (in German),

or Corollary 10.15 in:

T. Jech,

Set Theory - The Third Millennium Edition, Revised and Expanded, Springer Monogr. Math., Springer, Berlin, 2006 (corrected 4th printing).

However, the existence of a weakly inaccessible cardinal is unprovable from the axioms of ZFC.

On the other hand, we have from:

R. M. Solovay, ``Real-valued measurable cardinals'', pp. 397-428 in: D. Scott (ed.),

Axiomatic set theory, Proc. Sympos. Pure Math. (Univ. California, Los Angeles, CA, 1967), Vol. XIII, Part I, Amer. Math. Soc.: Providence, RI, 1987 (reprinted ed.)

that the existence of measurable cardinals in ZF is equiconsistent with the existence of measurable (resp., real-valued measurable) cardinals in ZFC. However, this doesn't answer the question I'm posing, as far as I can say (in particular, it occurs to me that Solovay's result doesn't cover the case of real-valued measurable cardinals in ZF, does it?).

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