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Let $K \subset \mathbb{R}$ be compact, and let $E$ be a separable Banach space. Further, let $(\Omega, \mathcal{F},\mathbb{P})$ a probability space. I would like to show that a certain a random function $$ f \colon \Omega\to C(K,E) $$ is measurable, where $C(K,E)$ is the space of continuous functions on $K$ to $E$, with the supremum norm. This means I have to show $f^{-1}(B) \in \mathcal{F}$ for all $B \in \mathcal{B}(C(K,E))$, but I do not know where to start, as I do not know exactly how Borel sets $B$ in $\mathcal{B}(C(K,E))$. Can anyone help me getting started?

Thanks

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  • $\begingroup$ what is C(K,E)? Do you mean continuous functions from K to E? $\endgroup$
    – Thomas Kojar
    Commented Feb 19, 2023 at 21:10
  • $\begingroup$ @ThomasKojar thanks, I have now changed the question so as to explain that $C(K,E)$ is the space of continuous functions on $K$ to $E$, with standard norm. $\endgroup$
    – Gerardo Mastracchio
    Commented Feb 19, 2023 at 21:14
  • $\begingroup$ In terms of measurability, we need to know the particular $f$. In terms of Borel sets in $C(K,E)$, this is a function space and so you need to equip it with some norm eg. supremum norm; then they will have the sup-norm topology. Whereas if you equipped it with L2 norm, the Borel sets will be different. $\endgroup$
    – Thomas Kojar
    Commented Feb 19, 2023 at 21:16
  • $\begingroup$ Also, this is a research question, really: the function $f$ I am considering comes from the solution to a Banach-valued dynamical system, and I am trying to understand how to prove measurability for the particular $f$ that I am after $\endgroup$
    – Gerardo Mastracchio
    Commented Feb 19, 2023 at 21:16
  • $\begingroup$ eg. see here math.stackexchange.com/questions/148349/… $\endgroup$
    – Thomas Kojar
    Commented Feb 19, 2023 at 21:20

1 Answer 1

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The Borel $\sigma$-algebra on the space $C(K,E)$ of continuous functions from a compact metrizable space $K$ to a separable metric space with the induced uniform metric is generated by the evaluation maps of the form $f\mapsto f(x)$. See, for example, Lemma 4.53 in Aliprantis & Border (2006).

Therefore, your random function is measurable if for every $x\in K$, the value at $x$ is a random variable.

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