Structure of generated sigma algbra

[Mike.B]

I am think what is the structure of generated $\sigma$-algebra. Let me make it specific. How to represent $\sigma(\mathscr{A})$, where $\mathscr{A}$ is an algebra. Can I use the elements of $\mathscr{A}$ to represent the element in $\sigma(\mathscr{A})$?

 

[Stephen Tashi]

That is an interesting question. We could make it even more specific. Is each $S \in \sigma(\mathscr{A})$ representable as $S = lim_{n \rightarrow \infty} ( \cup_{i=0}^n A_i )$ where $A_i \in \mathscr{A}$ ? (I don't claim to know the answer, but it seems like a good place to start.)