I understand that a random variable $X$ is a function from the sample space $\Omega$ to $\mathbb{R}$. What I fail to understand is why people when defining a new random variable $Y$, based on some function $g$ that operates on the output of $X$, use the notation $Y=g(X)$. I have seen that both in this site and also on the MIT lectures on probability by Prof. Tsitsiklis (here on 30:55 and onwards). I think this notation makes no sense at all. The domain of $g$ is $\mathbb{R}$, not some "set of functions from $\Omega$ to $\mathbb{R}$" (which is what the "set of random variables" is). In my view the correct notation ought to be:
$$ Y = g \circ X $$
where $\circ$ is function composition. Seen in this light $g \circ X$ is a function from $\Omega$ to $\mathbb{R}$ (hence, a legitimate random variable) and $g(x)$ (lowercase $x$) is a value in $\mathbb{R}$. Am I missing something?