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Let $(\mathcal{C}, \|\cdot\|)$ be a (non-locally compact) Banach space with Borel $\sigma$-algebra $\mathcal{B}$.

Given a probability measure $\mu : \mathcal{B}\rightarrow[0,1]$, I'm interested in regularity (''growth'') conditions on $\mu$ which guarantee that

$$\tag{1}\|\mu\|:=\int_{\mathcal{C}}\!\|x\|\,\mu(\mathrm{d}x) \, < \, \infty.$$

Clearly, probability measures with bounded support are of this form, but clearly there must be a reference where (Fernique-like) conditions of the form $(1)$ are studied in greater generality?

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    $\begingroup$ These are Radon measures of order 1 from the theory of Radonifying mappings due to Laurent Schwartz. Described in many sources, many in French, some in English. $\endgroup$
    – user95282
    Commented Jun 12, 2022 at 11:55
  • $\begingroup$ @user95282 Excellent, thank you! $\endgroup$
    – fsp-b
    Commented Jun 12, 2022 at 19:05

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