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My apologies for the title in advance - but I have a question about notation (I think). Let $X$ be a discrete random variable on the probability space $(\Omega,\mathscr F,\mathbb P)$ and let $g:\mathbb{R}\to\mathbb{R}$. It is easy to check that $Y=g(X)$ is a discrete random variable on $(\Omega,\mathscr F,\mathbb P)$. Now my books shows how to get the mass function of $Y$:

$\begin{align*} p_Y(y)=\mathbb P(Y=y)&=\mathbb P(g(X)=y) \\& =\mathbb P\left(X\in g^{-1}(y)\right)\\&=\sum_{x\in g^{-1}(y)}\mathbb P(X=x) \end{align*}$

I am a little bit hesitant on the following notation: $\mathbb P(g(X)=y) =\mathbb P\left(X\in g^{-1}(y)\right)$. Intuitively, I get it: the chance that the discrete variable $g(X)$ takes the value $y$ is the same chance that the discrete random variable $X$ takes a value that is an element of $g^{-1}(y)$. But is this true by definition or is it actually an elementary theorem?

So is there anything formal to say about the following;

$g(X)=y \iff X\in g^{-1}(y)$,

or should I just accept that it's something so fundamentally clear/logical, that I just have to work with it?

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    It's just a notation. – 2017-01-15
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    $$g^{-1}(y)=\{x\in\mathbb R\mid g(x)=y\}$$ – 2017-01-15
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    Oh ok. I'm disappointed and not personally convinced that there isn't more to say (I'm a beginning student in mathematics), but so be it – 2017-01-15
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    Personal convictions, if not supported by precise mathematical arguments, are not relevant here. – 2017-01-16

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