I am studying probability, specifically regular conditional distributions, and came across the notation $P[X\in dx]$. What does this mean? Here, $X$ is a random variable and $P$ is a probability measure.
I am also curious about $$\frac{P[Y\in dy\ |\ X=x]}{dy}.$$
Best Answer
If a random variable $X$ is given, then it induces the pushforward measure defines by
$$ E \mapsto \Bbb{P}( \{ \omega \in \Omega : X(\omega) \in E \}) = \Bbb{P}(X^{-1}(E)). $$
Mathematicians simply abbreviate this by $\Bbb{P}(X \in E)$ whenever no confusion arises. Replacing the particular choice of $E$ by a placeholder $\cdot$, we may symbolically write this pushforward measure by $\Bbb{P}(X \in \cdot)$.
If $X$ is real-valued, then $\Bbb{P}(X \in \cdot)$ defines a probability measure on $\Bbb{R}$. Now, recall a measure $\mu$ on $\Bbb{R}$ is often written symbolically as $\mu(dx)$, particularly in the context of integration where explicitly writing the variable on which integrands depend becomes important. Then the notation $\Bbb{P}(X \in dx)$ reduces to a particular case of this practice.
You can think that the symbolic notation $dx$ intuitively stands for any possible choices of infinitesimally small measurable sets. This practice is partially justified by the fact that if $\Bbb{P}(X \in \cdot)$ is a Borel measure on $\Bbb{R}$, then for any $f \in C_b(\Bbb{R})$,
$$ \int_{\Bbb{R}} f(x) \, \Bbb{P}(X \in dx) = \lim_{n\to\infty} \sum_{k=-\infty}^{\infty} f(x_k) \Bbb{P}(X \in [x_k, x_k + \Delta x) ), \quad \Delta x = \frac{1}{n} \text{ and } x_k = k \, \Delta x. $$