The monotone class theorem in my measure theory text book is stated as follows.
$\mathcal{C}$ is a family of sets. If $\mathcal{C}$ is an algebra, then $m(\mathcal{C}) = \sigma(\mathcal{C})$, where $m(\mathcal{C})$ resp. $\sigma(\mathcal{C})$ is the smallest monotone class resp. $\sigma-$algebra that contains $\mathcal{C}$.
And there are 2 related theorems below it:
If $\mathcal{C}$ is a family of sets satisfying $A \in \mathcal{C} \Rightarrow A^c \in m(\mathcal{C})$ and $A,B \in \mathcal{C} \Rightarrow A\cap B \in m(\mathcal{C})$, then $m(\mathcal{C}) = \sigma(\mathcal{C})$. ($A^c$ is the complement of $A$.)
If $\mathcal{C}$ is a family of sets satisfying $A \in \mathcal{C} \Rightarrow A^c \in m(\mathcal{C})$ and $A,B \in \mathcal{C} \Rightarrow A\cup B \in m(\mathcal{C})$, then $m(\mathcal{C}) = \sigma(\mathcal{C})$.
I wonder how to prove these 2 theorems. Is it possible to prove them from monotone class theorem? Or must they be proven from scratch, like the proof of the monotone class theorem, by proving that $m(\mathcal{C})$ is a $\sigma-$algebra.
Thanks in advance!