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

Question

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?

Answer 1

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 $x$.