I am self-studying a book on Measure Theory and Probability Theory by Lahiri and Athreya. I encountered the following definition of a Semi-Algebra.
Let $Ω$ be a nonempty set and let $P(Ω)$ be the power set of $Ω$. A class $C⊂P(Ω)$ is called a semialgebra if
(i) $A, B \in C \Rightarrow A \cap B \in C $
(ii) For any $A \in C$ there exists sets $B_1, B_2, ..., B_k \in C$ for some $1 \le k \lt \infty$, such that $B_i \cap B_j = \emptyset$ and $A^c = \bigcup_{i=1}^{k} B_i$ .
Following that, the book says that it can easily be shown that the smallest algebra containg $C$ is $\mathscr A =$ { $A : A = \bigcup_{i=1}^{k} B_i, B_i \in C$ for $i= 1, ... , k, k \lt \infty$ }
My problem is that I cannot seem to show the proof of the theorem. After searching for some other stackexchange questions, there is an additional assumption in the definition.
(iii) $\emptyset \in C$
or (iii') $ Ω \in C$
It seems like this solves the problem because I cannot seem to show that the algebra generated by the semi - algebra does not necessarily contain the empty set and the entire set $Ω$.
My question is, did the book have a mistake in its definition by ommiting an extra assumption or does the third assumption I just stated follow from the two above. If yes, how do you show that one of those I have mentioned are there?
