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

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.; Sikor …
Robert Furber's user avatar