A topological field is one with a topology that makes addition, multiplication, and non-zero inversion continuous. Real and Complex numbers with their standard topology are the obvious examples.
Given a Borel space $(X, \sigma_X)$ and a topological field with its corresponding Borel sigma algebra $(F, \sigma_F)$ I believe the set of all Borel measurable functions $f: X \to F$ forms a vector space.
I can't find a reference to confirm this and would appreciate feedback.
Sure. Define both addition and (scalar) multiplication pointwise:
$$(f+g)(x):= f(x)+g(x)$$ $$(kf)(x):= kf(x)$$
Now consider following functions:
$$A:F\times F\to F$$ $$A(x,y)=x+y$$ $$B_{k}:F\to F$$ $$B_{k}(x)=kx$$ $$C:X\to X\times X$$ $$C(x)=(x,x)$$
They are all continuous hence measurable. Now all you need to know is that
$$f+g = A\circ(f\times g)\circ C$$ $$kf = f\circ B_k$$
The last function is obviously measurable as a simple composition of measurable functions. $f+g$ is measurable as well because it is a composition of measurable functions as well. You just have to remember that the product of measurable functions $f\times g:X\times X\to F\times F$ is measurable.
In particular $f+g$ and $kf$ are well defined. I leave it as a simple exercise that it defines a vector space structure on the set of all measurable functions $X\to F$.