Help understanding set I am given the following, along with the following description:

Assume all functions have domain $ \mathbb R $. I will define a new concept. For every pair of functions $ f $ and $ g $, we define the set
$$ \Omega ^ g _ f = \{x \in \mathbb R : f ( x ) < g ( x ) \} \text . $$

My Qestion:
What exactly does this mean? I am being thrown off by the omega part. Why is $ g $ above $ f $? What exactly is this telling me and why is there the omega symbol? Is it just a random/convention variable name?
 A: I think you are overthinking this.
They are inventing a piece of notation, right then and there, and giving you the definition. That means you don't have to think about what each part means, at all. Just remember that from now on, each time you see an $\Omega$ with two functions attached to it, that's shorthand for a certain subset of the real numbers related to those two functions. Which subset? That's given by the right-hand side of the definition.
$\Omega_f^g$ is the subset of the real line given by where $g$ is larger than $f$. That's it.

Why is $g$ above $f$

Presumably to remind you of the fact that we're interested in the region where the graph of $g$ is above the graph of $f$. A mnemonic, and nothing of actual mathematical consequence. The two function names must appear somewhere, and that's as good a place as any.

why is there the omega symbol?

Presumably because the author thought it fitting for some reason. You need something, as just "$^g_f$" looks a bit strange. $\Omega$ is as good as any other symbol.

Is it just a random/convention variable name?

Yes, that is exactly what it is. Presumably, these kinds of sets, given by where one function is larger than another, is going to appear a lot in the following text. They wanted a shorter way to write it, so they made up some new notation on the spot for that purpose. They are kind enough to inform you of this new notation through this definition, so that you too can more compactly enjoy the theorems and proofs that follow.
