Dynkin system/Definition 2

$$\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern-15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}} }$$ $$\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}$$ $$\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }$$ Given a set $X$ and a family of subsets of $X$ we denote $\mathcal{D}\subseteq\mathcal{P}(X)$ is a Dynkin system on $X$ if:

• $X\in\mathcal{D}$
• $\forall A,B\in\mathcal{D}[B\subseteq A\implies A-B\in\mathcal{D}]$
• Given a sequence $(A_n)_{n=1}^\infty\subseteq\mathcal{D}$ that is increasing and has $\lim_{n\rightarrow\infty}(A_n)=A$ we have $A\in\mathcal{D}$