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Before diving into the Stanford CS229 Machine Learning notes online, I decided to go through the course's notes on probability review and had a few questions.

In section 2.2, it states

A probability mass function (PMF) is a function $p_x : \Omega \rightarrow \mathbb{R}$ such that $p_X(x) = P(X=x)$

However, wouldn't a PMF defined this way be a function defined from $\mathcal{F} \rightarrow \mathbb{R}$ not $\Omega \rightarrow \mathbb{R}$, since $X = x$ represents the set $\{\omega : X(\omega) = k\}$, which is itself a member of $\mathcal{F}$?

Also, another question. In section 2.5, it states

Suppose that $g(x) = 1\{x \in A\}$ for some subset $A \subseteq \Omega$.

I don't understand what $x \in A$ means. $x$ here seems to be some value the random variable $X$ can take, i.e. $X = x$. What does it mean for a random variable value to be a member in a subset of the sample space $\Omega$?

My guess is that $x$ here is shorthand for the set $\{\omega : X(\omega) = k\}$ which is itself a subset of $\Omega$.

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    $\begingroup$ +1 The notation in that document is shockingly sloppy. You can readily resolve your questions, though, by consulting Wikipedia. See, e.g., en.wikipedia.org/wiki/Probability_mass_function. $\endgroup$
    – whuber ♦
    Commented Jan 18, 2019 at 17:19
  • $\begingroup$ How are $\Omega$ and $\mathcal{F}$ defined in the first quote? A pmf associates to every possible realisation of the random variable $X$ in $\Omega$ a probability between $0$ and $1$ so I see no issue with this definition. And $\mathbb{1}A(x)$ is the indicator function, taking value $0$ when $x\not\in A$ and $1$ when $x\in A$. Which is a notion unrelated with random variables. $\endgroup$
    – Xi'an
    Commented Jan 18, 2019 at 20:23
  • $\begingroup$ ... (assuming $\Omega$ is the set of all possible realisations and not the standard origin set for the random variable $X=X(\omega)$...) $\endgroup$
    – Xi'an
    Commented Jan 18, 2019 at 20:29

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