Density character of a metric space is an Ulam number

Question

I am reading this paper and I came across the following sentence:

Throughout the paper we silently assume [...] that the density character (i.e. the minimum cardinality of a dense subset) of every metric space is an Ulam number. This guarantees that every finite positive Borel measure is tight (even Radon when the space is complete), is concentrated on some $\sigma$-compact subset and the support of this measure is separable.

I have essentially no background in logic or descriptive set-theory. Could you please quickly tell me what this silent assumption amounts to? What is an Ulam number? I did some googling but got no serious answer. Furthermore, I suppose this is consistent with ZF(C?) but I have really no idea of what we are talking about, so I hope you can shed some light on this.

Answer 1

Two mutually equivalent definitions of an Ulam number are given in Section Weak inaccessibility of real-valued measurable cardinals.

Added in response to a comment by the OP: First here, the first of the two referenced definitions can be restated, more compactly, as follows: a cardinal $\alpha$ is an Ulam number iff for every set $X$ of cardinality $\le\alpha$ there is no nonzero finite nonatomic outer measure $\mu$ over $X$ such that all subsets of $X$ are $\mu$-measurable.

As for how to regard the condition "that the density character (i.e. the minimum cardinality of a dense subset) of every metric space is an Ulam number", one may note the following:

In the same section of the referenced Wikipedia article, we find:

a cardinal that is not an Ulam number is weakly inaccessible

Further, in the first paragraph of Section Models and consistency, we see

ZF together with "there exists a weakly inaccessible cardinal" implies that ZFC is consistent. Therefore, [weakly] inaccessible cardinals are a type of large cardinal.

Further yet, in the first paragraph of article Large cardinal, we find:

The proposition that such cardinals exist cannot be proved in the most common axiomatization of set theory, namely ZFC

Thus, the existence of a metric space whose density character is not an Ulam number cannot be proved in ZFC. So, the condition that

the density character (i.e. the minimum cardinality of a dense subset) of every metric space is an Ulam number

might be considered a mild restriction -- say, in the sense that the density character of any metric space that one may encounter "in practice" will be an Ulam number.

Answer 2

An Ulam number is some authors' term for what is called "a cardinal smaller than the first real-valued measurable cardinal" by set theorists in general, or "a measure-free cardinal" by D. H. Fremlin (I also prefer this terminology). In Ulam's original paper he called them "unmeasurable" (but in German), this should no longer be used because it contradicts the modern terminology for measurable cardinals. For the sake of consistency I will carry on using the "Ulam number" terminology. A short way of defining them is to say that it is a cardinal $\kappa$ for which every probability measure on the measurable space $(\kappa, \mathcal{P}(\kappa))$ arises from an $\ell^1$ function. (I don't know what the Wikipedia article is driving at with that "an outer measure such that each set is measurable" stuff -- that's just what most people would call a measure defined on all subsets. Maybe I'll change it later.)

It is stated in the linked paper that if the density character of a metric space $X$ is an Ulam number, then every Borel measure on $X$ is tight. This is false (though I don't think they ever use it for an incomplete metric space later, so the paper looks OK). If we take $X \subseteq [0,1]$ to be a set with outer Lebesgue measure 1 and inner Lebesgue measure 0, metrize it as a subspace, and restrict Lebesgue measure to it (defining $\mu(X) = 1$, I can explain this more if you've never seen this type of construction before) then we obtain a separable (incomplete) metric space $X$ with a non-tight probability measure $\mu$ on it.

The true facts are:

1. If $X$ is metrizable and has density character an Ulam number, then for each probability measure $\mu$, there exists a separable subspace $Y \subseteq X$ with $\mu(Y) = 1$.
2. If $X$ is completely metrizable and has density character an Ulam number, then every probability measure is tight and Radon.

The proofs can be found in the appendix to the first edition of Billingsley's "Convergence of Probability Measures" (as Iosif Pinelis discovered, this important fact was removed by the Commissar for Dumbing Down in later editions).

Moreover, although the smallest non-Ulam number (i.e. the first real-valued measurable cardinal) must be weakly inaccessible, it could still (consistently) be less than or equal to $2^{\aleph_0}$, as long as "ZFC + there exists a measurable cardinal" is consistent in the first place. This was proved by Solovay. The density character of $\ell^\infty$ in its norm topology is $2^{\aleph_0}$, so the authors of the paper need their set-theoretic assumption at this point (as they clearly state).