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I have seen the Uniform Distribution/a uniform random variable for some interval in $mathbb{R}$. For example $U(a,b)$ has probability density function $frac{1}{b-a}$ (noting this is the 'volume' of the interval)

My question : Is there a general Uniform Distribution/a general uniform random variable for a set $Omega$ which is a subset of $mathbb{R}^{n}$ , I assume if so the p.d.f will be the volume of $Omega$ ?

If this exists could anyone tell me some of its properties like its mean, its variance, its p.d.f, /any other interesting things about it?

If this distribution (or random variable) has not been extensively studied and the above properties can not be provided, then can we make $Omega$ 'nice enough' so that it does?

Its PDF would be $1/mathrm{Vol}Omega$ for $x in Omega$ and $0$ otherwise.

Its mean, variance, and other properties would all depend on your choice of $Omega$. Without a fixed $Omega$, you couldn't say anything interesting about mean, variance, etc.

This is because the PDF is still defined on all of $R^n$ (it's just $0$). It's not really uniform in that $Omega$ can be highly irregular, not connected, etc. So what you're asking is "given an arbitrary set $Omega subset R^n$, find its center of mass, variance, and other properties". It's pretty easy to see that you can't say much without a specific set.