Give an example of two $\sigma$ algebras in a set $X$ whose union is not an algebra.
I've considered the sets $\{A|\text{A is countable or $A^c$ is countable}\}\subset2^\mathbb{R}$, which is a $\sigma$ algebra. I also tried to generated a $\sigma$ algebra from a collection of $\sigma$ algebra, but I've been unfruitful. So far I know $2^X,\{\emptyset, X\},$ and the measurable sets $\mathcal{L}$ are $\sigma$ algebras, but they haven't helped me too much.
Thank you!
Best Answer
The trivial $\sigma$-algebra on $X$ is $\{\emptyset, X\}$. The next simplest $\sigma$-algebras on $X$ are $\{\emptyset, A, A^c, X\}$ where $A \in \mathcal{P}(X)\setminus\{\emptyset, X\}$. Can you construct a counterexample using such $\sigma$-algebras?