# Ultraproduct of a Function versus Functions of Ultraproducts

Let $$G$$ be a group (or even a set for our purposes here) and consider functions from $$G$$ to $$\mathbb{R}$$. Now after choosing a non-principal ultrafilter of $$\mathbb{N}$$, we can construct ultrapowers $$^*G$$ and $$^*\mathbb{R}$$. The usual metric on $$\mathbb{R}$$ extends to an ultrametric on $$^*\mathbb{R}$$

Suppose we have the space $$C=\{ f:^*G \to ^*\mathbb{R}\}$$. Now consider subspaces $$C_1 = \{ f \in C: \|f\| \text{ is bounded}\}$$ $$C_0 = \{f \in C: \|f\| \text{ is infinitesimal} \}$$ Here $$\|f\|$$ is the infinity norm on the function, or the supremum of $$f$$ over all inputs, and is now an element of $$^*\mathbb{R}$$.

Note that $$C_0 \subset C_1$$. Consider the quotient $$C_1/C_0$$ which is the space of equivalence classes of bounded functions modulo infinitesimals.

I have a couple of questions:

Can $$C_1/C_0$$ simply be treated as the space of functions $$\{g: ^*G \to \mathbb{R}\}$$ since $$^*\mathbb{R}_{bounded}/^*\mathbb{R}_{infinitesimal} \equiv \mathbb{R}$$?

Can the correspondence be given as follows: for $$f \in C_1$$, define $$\tilde{f} \in C_1/C_0$$ by $$\tilde{f}(g)=st(f(g))$$, where $$st$$ is the standard part function which takes a finite hyperreal and returns its closest real number?

Now suppose we further restrict ourselves to functions that are internal, or in other words, themselves arise as ultraproducts of functions from $$G$$ to $$\mathbb{R}$$. Now we define $$\tilde{C}$$, $$\tilde{C}_1$$ and $$\tilde{C}_2$$ the same way $$\tilde{C}=\{ f:^*G \to ^*\mathbb{R} \text{ is internal}\}$$ $$\tilde{C}_1 = \{ f \in \tilde{C}: \|f\| \text{ is bounded}\}$$ $$\tilde{C}_0 = \{f \in \tilde{C}: \|f\| \text{ is infinitesimal} \}$$ Again, consider $$\tilde{C}_1/\tilde{C}_0$$.

Assuming that the elements of $$\tilde{C}_1/\tilde{C}_0$$ can also be thought of as functions from $$^*G$$ to $$\mathbb{R}$$, is there any way in which the fact that they are internal reveals itself here?

• Eh, I don't think the supermum exists. It certainly doesn't transfer, being a second order property. – PyRulez Feb 13 at 14:43