Notations:

• Set theoretic notations: union : $A \sqcup B$; intersection : $A\sqcap B$; disjoint union is denoted: $A \mho B$ (\mho in TeX settings):$\mho_{i = 1}^{n} A_i$ ;
• Orders: less than or equal to: $a \leqslant b$; greater than or equal to: $a \geqslant b$;
• Standard sets for numbers: $\mathbb{N} = \{0, 1, 2, \dots\}$; $\mathbb{Z}^{+}$: positive integers, i.e. $x\in \mathbb{Z}, x > 0$; $\mathbb{R}^{+}$: positive reals, i.e. the set of reals $x>0$;
• Topology : $B_{r}(x)$: open ball of radius $r>0$ around $x$; $B_{r]}(x)$: closed ball of radius $r>0$ around $x$;
• Measure and Integration : $\mu[A]$: the measure of the measurable set $A$; if $T: (X, \mu) \to Y$, $T_{\#}\mu$ is the measure image of $\mu$, i.e. its pushforward, defined as $T_{\#}\mu[B] = \mu[T^{-1}(B)]$. A transformation $T: (X, \mu) \to (Y, \nu)$ is said to be measure preserving if $\nu = T_{\#}\mu$.
• Abstract Algebra: Group of units $U(R)$ where $R$ is a ring; zero divisors $ZD(R)$; normal subgroup $H \trianglelefteq G$;