pr.probability - Existence of probability measure defined on all subsets

Here is some elaboration on Joel David Hamkins's answer.

A (two-valued) measurable cardinal is an uncountable cardinal $kappa$ such that there is a continuous $({<kappa})$-additive ${0,1}$-valued probability measure $mu$ defined on all subsets of $kappa$. To say that $mu$ is $({<kappa})$-additive means that if $(X_i)_{i in I}$ is a family of pairwise disjoint subsets of $kappa$ with $|I|<kappa$, then $muleft(bigcup_{i in I} X_iright) = sum_{iin I} mu(X_i)$. Since $mu$ can only take values $0$ and $1$, this is equivalent to saying that (a) at most one of the $X_i$ can have measure $1$, and (b) if they all have measure $0$ then so does $bigcup_{i in I} X_i$. (Some people say $kappa$-additive instead of $({<kappa})$-additive, but I prefer to use $kappa$-additive to mean the above for families with index set of size equal to $kappa$.)

The existence of a continuous countably additive ${0,1}$-valued probability measure $mu$ defined on all subsets of a set $S$ implies the existence of a measurable cardinal. Indeed, I claim that if $kappa geq aleph_1$ is the smallest cardinal such that $mu$ is not $kappa$-additive, then $kappa$ is a measurable cardinal. To see this, let $(X_i)_{i<kappa}$ be a family of pairwise disjoint measure $0$ sets such that $bigcup_{i<kappa} X_i$ has measure $1$ (i.e. the family contradicts $kappa$-additivity in the only possible way). Defining $bar{mu}(I) = muleft(bigcup_{i in I} X_iright)$ for every $I subseteq kappa$, we obtain a $({<kappa})$-additive ${0,1}$-valued probability measure $barmu$ defined on all subsets of $kappa$.

So the existence of a continuous countably additive ${0,1}$-valued measure defined on all subsets of a set $S$ is exactly equivalent to the existence of a measurable cardinal. However, since a probability measure is also allowed to take values strictly between $0$ and $1$, this is not quite equivalent to the statement you asked about.

By analogy with the above, a real-valued measurable cardinal is an uncountable cardinal $kappa$ such that there is a continuous $({<kappa})$-additive probability measure defined on all subsets of $kappa$. The existence of a real-valued measurable cardinal is equivalent to your statement by a variation of the trick used above.

In 1930, Ulam (Zur Masstheorie in der allgemeinen Mengenlehre, Fund. Math. 16) showed that if $kappa$ is real-valued measurable then $kappa geq 2^{aleph_0}$, and that if $kappa > 2^{aleph_0}$ is real-valued measurable then $kappa$ is in fact measurable (with a possibly different measure). Ulam also showed that successor cardinals like $aleph_1$ cannot be real-valued measurable.

In the 1960's, Solovay (MR290961) finally resolved the boundary case. He showed that if $kappa = 2^{aleph_0}$ is real-valued measurable then there is an inner model (namely $L[I]$ where $I$ is the ideal of null sets) wherein $kappa$ is still real-valued measurable and GCH holds, therefore $kappa$ is measurable in that inner model by Ulam's earlier results. While this doesn't mean that the existence of a real-valued measurable cardinal and the existence of a measurable cardinal are equivalent, it shows that the two statements are equiconsistent over ZFC.

Using forcing (and another result of Ulam), Solovay also showed that if there is a model with a measurable cardinal then there is a model in which the Lebesgue measure on $[0,1]$ can be extended to a probability measure defined on all subsets of $[0,1]$.