I am reading [this](https://arxiv.org/pdf/1303.5664.pdf) 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. 



Density character of a metric space is an Ulam number
Two mutually equivalent definitions of an Ulam number are given in Section [Weak inaccessibility of real-valued measurable cardinals](https://en.wikipedia.org/wiki/Measurable_cardinal#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
> 
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> 


Further, in the first paragraph of Section [Models and consistency](https://en.wikipedia.org/wiki/Inaccessible_cardinal#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. 
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Further yet, in the first paragraph of article [Large cardinal](https://en.wikipedia.org/wiki/Large_cardinal), we find: 



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


