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 $f \in C_b(\Bbb{R})$,
$$ \int_{\Bbb{R}} f(x) \, \Bbb{P}(X \in dx) = \lim_{n\to\infty} \sum_{k=-\infty}^{\infty} f(x_k) \Bbb{P}(X \in [x_k, x_k + \Delta x) ), \quad \Delta x = \frac{1}{n} \text{ and } x_k = k \, \Delta x. $$
As a set,
$$xHx^{-1}=\{xhx^{-1}\mid h\in H \}.$$
It is a nice exercise to show that
$$\bigcap_{x\in G}xHx^{-1}$$
is a normal subgroup of $G$ (by construction essentially) and that it is the largest normal subgroup of $G$ contained in $H$. This thing is called the core of $H$.
Best Answer
Generally $a \mid b$ means $a$ divides $b$ where $a,b\ne 0$ are integers. Since $\frac{b}{a}$ is a rational number it is sometime cumbersome to say that $\frac{b}{a}$ is an integer even though it means the same thing. Even we can write in words that " $b$ is divisible by $a$" but $a \mid b$ convention is lot more easy to follow.