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$\DeclareMathOperator\Fin{Fin}$I am trying to understand the notion of "liftings" of Loeb measurable functions to internal, internally measurable functions, and its connection to non-standard hulls of Banach spaces. Apologies for not being too precise with the details below, since that would clutter up the space. The reference is "Nonstandard analysis- Theory and applications" by Arkeryd, Cutland and Henson.

A toy situation is as follows: Suppose $(\Omega,\mathcal{A},{}^*\mu)$ is an internal probability space. That is $\Omega$ is an internal set with a finitely additive measure on an internal algebra of subsets taking values in ${}^*\mathbb{R}$. This can be used to construct a Loeb measure on $\Omega$ which is $\sigma$-additive and takes real values. Given a measurable (using Loeb measure on $\Omega$ and the standard Lebesgue measure on $\mathbb{R}$) function $f\colon\Omega \to \mathbb{R}$, there is a notion of a (unipedal) lifting of $f$ to get an internal function $F\colon\Omega \to {}^*\mathbb{R}$ such that $F$ and $f$ agree up to infinitesimals up to a Loeb null set. This lifting is not limited to functions to $\mathbb{R}$ but also more general Radon measure spaces $X$. That is, a Loeb measurable $f\colon\Omega \to X$ can be lifted to an internal $F\colon\Omega \to {}^*X$ which agrees with $f$ up to standard part.

Meanwhile, given an internal Banach (or even normed linear) space $V$, we can consider its non-standard hull which is $\hat{V}=\Fin(V)/V_0$ where $\Fin(V)$ comprises elements of finite norm, while $V_0$ comprises elements of infinitesimal norm. This is a standard Banach space. My question is: can a Loeb measurable function $f\colon\Omega \to \hat{V}$ be lifted to an internal function $F\colon\Omega \to V$ that takes values in $\Fin(V)$ (assuming $f$ is bounded, say) and agrees with $f$ up to $V_0$ (think of it as our standard part function)?

As I understand it, $f\colon\Omega \to \hat{V}$ can be lifted to an internal $F\colon\Omega \to {}^*\hat{V}$, at least by what I stated earlier. Here however, I have a more specific internal space and a standard part map given by this non-standard hull construction.

Am I confusing the two notions? At least at the level of $\mathbb{R}$ it all comes together well, since $\widehat{{}^*\mathbb{R}}={}^*\mathbb{R}_{\mathrm{b}}/{}^*\mathbb{R}_{\inf} = \mathbb{R}$. But for a general (standard) Banach space $V$, what is the relationship between $\widehat{{}^*V}$ and $V$, and for an internal space, is there a relationship between ${}^*\hat{V}$ and $V$?

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  • $\begingroup$ I'm not completely sure if this is what you're asking, but $\widehat{{}^*V}$ will be a strict superspace of $V$ whenever $V$ is an infinite dimensional Banach space. They are closely related in terms of their continuous first-order theories, but this is perhaps a bit subtler than what you're looking for. $\endgroup$ – James Hanson Dec 21 '20 at 17:44
  • $\begingroup$ @James Hanson: Thanks, that is helpful to know. My broader question, however, is whether Loeb measurable functions to $\hat{V}$ can be lifted to internal measurable functions to $V$ (as opposed to just ${}^*\hat{V}$). $\endgroup$ – BharatRam Dec 23 '20 at 5:21
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Yes, $f:\Omega \rightarrow \hat{V}$ has a lifting to some function $F : \Omega \rightarrow V$. This is shown in section 4 of the paper "Lifting theorems in nonstandard measure theory", D. Ross, 1990: https://www.ams.org/journals/proc/1990-109-03/S0002-9939-1990-1019753-0/S0002-9939-1990-1019753-0.pdf

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