# Loeb measures and non-standard hull of Banach spaces

$$\DeclareMathOperator\Fin{Fin}$$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 $$V$$?

• 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. – James Hanson Dec 21 '20 at 17:44
• @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 ${}^*\hat{V}$). – BharatRam Dec 23 '20 at 5:21

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