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I am readin Ross's "A first course in probability" and I got to the chapter that talks about random variables.

I am tryng to understand the exact meaning of something of the form $g(X)$ where g is a real valued function and X is a r.v.

I can't figure the "form" of g, that is, I don't understand what to write in $g: ?\to ??$.

I saw an example that made me think $g:\mathbb{R}\to\mathbb{R}$, is this correct ?

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    You are right, $g$ would be a function from the reals to the reals, although it could be also on other sets and to other sets. But if $X$ is itself from $\Omega \to \mathbb{R}$, $g$ would need to be from $\mathbb{R}$.2012-04-23

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Some textbooks have the following framework. If $(\Omega,\mathscr{M},\mu)$ is a probability space and $X:\Omega\to\mathbb{R}$ is a r.v. (i.e., $\mu$-measurable function) and $g:\mathbb{R}\to\mathbb{R}$ (e.g. a continuous function), then $g(X):\Omega\to\mathbb{R}$ is a r.v. which is attained by the composition of $X$ and $g$, i.e. $g\circ X$.

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    Since our interest lies in the $\mu$-measurability of certain sets obtained as preimages by the function, it is quite natural to refer to the structure. It is also a means to avoid repeating the same phrase 'an element of the underlying $\sigma$-algebra $\mathscr{M}$', where $\mathscr{M}$ is indeed the collection of $\mu$-measurable sets. If everything is clear from the context, you may leave the measure out, yet I do argue that it is a quite standard notion to have it there, especially when you have multiple $\sigma$-algebras defined on $\Omega$...2012-04-23