In Mathematical finance (books; Mathematics of financial markets by robert J.elliot, Mathematic of arbitrage by freddy delbaen, Walter schachermayor), the following spaces are used for various things, $L^0(\Omega, \cal F,p, \mathbb{R}^n)$, $L^1(\Omega,\cal F,p,\mathbb{R}^n)$ and $L^\infty(\Omega,\cal F,p,\mathbb{R}^n)$.
the following three from measure theory (book Measure integrale and probability by Marek Capinski) and wiki https://en.wikipedia.org/wiki/Lp_space, I don't feel that I have issue understanding the following What does Lebesgue measure space look like?
$L^0$ representing the class of equivalent measurable spaces. which I understand.
$L^1$ representing the class of euivalent measurable space which are absolutely integrable and is finite.
$L^\infty$ representing the class of essentially bounded functions.
$L^0(\Omega,\cal F,p,\mathbb{R}^n)$-represents the equivalent measurable functions, in finance books this is used to denote the set of contingent claims. from here I loss it what does the elements of the set (functional vector space) looks like.
To the me the element of $f \in L^0(\Omega,\cal F,p,\mathbb{R}^n)$ looks something like this:
$f:\Omega \times T \rightarrow \mathbb{R}^n$, reasoning being that is $f(\omega, t) \in \mathbb{R}^n$ represent the event and t the time, being an element of $\mathbb{R}^n$ represent the discount gains process for T the terminal Gain. I know I am wrong in my understanding here, because the rest of the theories in the finance books doesnt make sense, can someone please explain what the elements of $L^0, L^1$ and $L^\infty$ look like.
The thing that confuse me is what does the domain and range of $f$ look like, like how are elements of $\omega \in \Omega$, $ A \in \cal F_t$, $t \in T$, $p$ and $\mathbb{R}^n$ used?
Thanks in advance