Probability distribution and probability measure are synonyms.
$[X=a]=X^{-1}(\{a\})=\{\omega\in\Omega\mid X(\omega)=a\}$ hence $P(X=a)=P(X^{-1}(\{a\}))$.
The distribution of the random variable $X:\Omega\to\mathbb R$ is the unique probability measure $\mu$ on $\mathcal B(\mathbb R)$ defined by $\mu(B)=P(X\in B)$ for every $B$ in $\mathcal B(\mathbb R)$, where $[X\in B]=X^{-1}(B)=\{\omega\in\Omega\mid X(\omega)\in B\}$.
$[a,b] \times [c,d]$, I think, would be recognized as the set you mean.
In my opinion, writing $H = \{(x,y)| (x,y) \in [a,b] \times [c,d]\}$ is redundant. If $(x,y)$ is in $[a,b] \times [c,d]$ then why the heck don't you your just write $H = [a,b]\times [c,d]$????? After all if you wrote $H = \{x|x \in \mathbb R\}$ that'd be seen as ridiculous; $H = \mathbb R$ fercripesake!
The only reason to put it in set notation is for clarity/definition as $[a,b]\times [c,d]$ might not be clear in meaning or might not be known to a novice. In which case $H = \{(x,y)| (x,y) \in [a,b] \times [c,d]\}$ does nothing to add to the clarity.
Why not just write $H = \{(x,y)| a \le x \le b; c \le y \le d\}$. That's perfectly clear and legit. Or $H = [a,b]\times [c,d] = \{(x,y)| a \le x \le b; c \le y \le d\}$ can be seen as a definition.
But to answer your question $H = [a,b]\times[c,d] := \{(x,y)| a \le x \le b; c \le y \le d\}$ is acceptable and standard notation.
Best Answer
$\mathbb K$ can denote a generic field (Körper in German),
though it would have been good if the Wikipedia article had defined it.
When there is a function, $\to$ indicates what domain is mapped to what codomain,
whereas $\mapsto$ indicates where it takes a particular element.
For example, $\det:M_n(\mathbb K)\to \mathbb K$, and $\det:I\mapsto1$.