Reference: If $X$ is metrizable, then $X$ is realcompact iff $|X|$ is non-measurable

Question

Note: What I call a measurable cardinal seems to be non-standard among set theorists, and should be called a $\sigma$-measurable cardinal.

I know that a discrete space is realcompact iff its non-measurable and I've been able to prove that the same holds for metrizable spaces here using the following result of Hirata.

Theorem. (corollary 2.3) Let $X$ be a semi-stratifiable space with $e(X) = \kappa$, where $\text{cf}(\kappa) > \omega$. Assume that $\tau^\omega < \kappa$ for each $\tau < \kappa$. Then $X$ has a closed discrete subset of size $\kappa$.

My proof was improvised however, could someone provide me with a reference to a theorem that says a metrizable space is realcompact if and only if it's of non-measurable cardinality?

Answer 1

You can consult Problem 8.5.13 in Engelking's General Topology. It deals with Dieudonné complete spaces (Tychonoff spaces that have a complete uniformity). Part (d) shows that every paracompact space is Dieudonné complete (the family of all open covers forms a uniformity that is complete), hence so is every metrizable space. Part (h) shows that a Dieudonné complete space is realcompact iff every closed discrete subspace is realcompact (iff every closed discrete subspace is of non-$\sigma$-measurable cardinality) with a reference to T. Shirota, A class of topological spaces, Osaka Math. J. 4, 23-40 (1952).

So this gives you the if and only if you seek: a metrizable space is realcompact iff every closed discrete subset is of non-$\sigma$-measurable cardinality, that is, of cardinality less than the first measurable cardinal. This implies that such a space is realcompact iff its weight is smaller than the first measurable cardinal, and since $|X|\le w(X)^{\aleph_0}$ if $X$ is metrizable you get the result you want. Theorem 4.1.15 in Engelking's book shows that all familiar global cardinal functions, in particular weight and extent, coincide on metrizable spaces.

Answer 2

Here is some information on the history of the result, which was actually proven before Shirota's 1952 theorem. It was proved in 1948 by Marczewski and Sikorski as Theorem VI in:

Marczewski, E.; Sikorski, R., Measures in non-separable metric spaces, Colloq. Math. 1, 133-139 (1948). ZBL0037.32201.

Their statement is that if $X$ is metrizable and $|X|$ is less than the first measurable cardinal, then every 2-valued Borel measure on $X$ is a Dirac measure (of some point or other in $X$). The converse is not directly stated, but is trivial - if $|X|$ is greater than or equal to the first measurable cardinal, it admits a countably-additive measure $\mu : \mathcal{P}(X) \rightarrow \{0,1\}$ vanishing on singletons, which can be restricted to a Borel measure, which, since it vanishes on singletons, is not the Dirac measure of any point.

Of course, to relate this to the usual definition of realcompact space, you need to use Hewitt's Theorem 16 from:

Hewitt, Edwin, Linear functionals on spaces of continuous functions, Fundam. Math. 37, 161-189 (1950). ZBL0040.06401.

which states that a completely regular space is realcompact iff every 2-valued Baire measure on it is a Dirac measure. (On a metrizable space, the Baire and Borel sets (in the relevant sense) are the same because for every closed set there is a real-valued continuous function vanishing exactly on that set.)

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Beware that back then they used an older terminology for measurable cardinals, where a cardinal $\kappa$ has "two-valued measure zero" iff it admits no countably-additive measure $\mu : \mathcal{P}(\kappa) \rightarrow \{0,1\}$ vanishing on singletons (equivalently, strictly smaller than the first measurable cardinal) otherwise it is "two-valued measurable" (in modern terminology, greater than or equal to the first measurable cardinal). To avoid confusion, no new publications should use the old terminology, even though it really is countably-additive measures that matter here (so one doesn't care about measurable cardinals larger than the first).