I have just started to study measure theory and I had been given the following definition of semi-algebra:
Let $\Omega$ be a nonempty set and let $\mathcal{P}(\Omega)$ be the power set of $\Omega$. A class $\mathcal{C}\subseteq\mathcal{P}(\Omega)$ is called a semi-algebra if
- $A,B\in\mathcal{C}$ implies that $A\cap B\in\mathcal{C}$.
- For any $A\in\mathcal{C}$, there exist sets $B_{1},B_{2},\ldots,B_{k}\in\mathcal{C}$, for some $1\leq k <\infty$ s.t. $B_{i}\cap B_{j} = \varnothing$ for $i\neq j$ and $A^{c} = \bigcup_{i=1}^{n}B_{i}$
My question is: what is the intuition underlying this definition?
At first glance, I am not able to understand the motivation to it.
Any contribution is appreciated.