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If a probability measure $\mathbb{P}$ is a function from the event space $\mathcal{F}$ to $[0, 1]$, that is, $\mathbb{P}: \mathcal{F} \rightarrow [0, 1]$, then why can we sometimes write $\mathbb{P}(X=x)$, where $X$ is a random variable, to indicate the probability that the random variable $X$ takes on the (arbitrary) value $x$, given also that a random variable is a function $\Omega \rightarrow \mathbb{R}$ and an event is a subset of $\Omega$, so an event is not a function (like a random variable)? What does the notation $\mathbb{P}(X=x)$ actually mean?

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    $\begingroup$ It is a convenience and should be written $P\{ \omega | X(\omega) = x \}$. Another notation is $P[X=x]$. $\endgroup$ – copper.hat Dec 22 '19 at 19:13
  • $\begingroup$ @copper.hat Feel free to provide a formal answer and I will at least upvote it. $\endgroup$ – nbro Dec 22 '19 at 19:14
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Let $(\Omega,\mathcal A,\Bbb P)$ be a probability space, meaning that $\Omega$ is the sample space, $\mathcal A$ a $\sigma$-algebra of events on $\Omega$ and $\Bbb P$ a probability measure. Whenever you have a random variable, that is a measurable function $X\colon\Omega\to [0,1]$ you can consider the pushforward measure $X_\ast\Bbb P$ of $\Bbb P$ through $X$, defined by $$X_\ast\Bbb P(A)=\Bbb P\left(X^{-1}(A)\right),\quad\text{for }A\in\Bbb R.$$

When we write $\Bbb P(X=x)$ what we mean is $X_\ast\Bbb P(\{x\})$, that is $\Bbb P\left(\left\{\omega\in\Omega\mid X(\omega)=x\right\}\right)$, it's just a shorthand, but the meaning should be clear if you think about $\Bbb P(X=x)$ as telling intuitively what's the probability that $X$ is equal to $x$.

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  • $\begingroup$ So, when we write $p(x)$, $x$ is an event and we actually mean $p(X=x)$, which actually means what you wrote in your answer? $\endgroup$ – nbro Dec 22 '19 at 19:52
  • $\begingroup$ $x\in\Bbb R$, $X^{-1}(x)$ is an event in $\mathcal A$ $\endgroup$ – Alessandro Codenotti Dec 22 '19 at 19:53
  • $\begingroup$ Ok, let me put it in another way. When we write p(x), we mean $p(X^{-1}(x))$? $\endgroup$ – nbro Dec 22 '19 at 19:55
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    $\begingroup$ $\Bbb P(X=x)$ means $\Bbb P(X^{-1}(x))$, $\Bbb P(x)$ looks weird and wrong to me, since $x$ is not in the domain of $\Bbb P$ $\endgroup$ – Alessandro Codenotti Dec 22 '19 at 19:56
  • $\begingroup$ Yes, but people (e.g. in machine learning or statistics) sometimes write p(x) or P(x) or $\mathbb{P}(x)$ or some variation. In those cases, do they mean $\mathbb{P}(X^{-1}(x))$? $\endgroup$ – nbro Dec 22 '19 at 19:57

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