# Why can we use the notation $\mathbb{P}(X=x)$ when $\mathbb{P}$ is a probability measure?

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?

• It is a convenience and should be written $P\{ \omega | X(\omega) = x \}$. Another notation is $P[X=x]$. – copper.hat Dec 22 '19 at 19:13
• @copper.hat Feel free to provide a formal answer and I will at least upvote it. – nbro Dec 22 '19 at 19:14

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 $$x$$.
• 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? – nbro Dec 22 '19 at 19:52
• $x\in\Bbb R$, $X^{-1}(x)$ is an event in $\mathcal A$ – Alessandro Codenotti Dec 22 '19 at 19:53
• Ok, let me put it in another way. When we write p(x), we mean $p(X^{-1}(x))$? – nbro Dec 22 '19 at 19:55
• $\Bbb P(X=x)$ means $\Bbb P(X^{-1}(x))$, $\Bbb P(x)$ looks weird and wrong to me, since $x$ is not in the domain of $\Bbb P$ – Alessandro Codenotti Dec 22 '19 at 19:56
• 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 $\mathbb{P}(X^{-1}(x))$? – nbro Dec 22 '19 at 19:57