Here is another very basic analysis problem but that puzzles me:
Find an example of set $X$ and its two $\sigma$-algebras $\mathscr A_1$ and $\mathscr A_2$, such that $\mathscr A_1 \cup \mathscr A_2$ is not $\sigma$-algebra.
To me at least, this question looks counter-intuitive since the union of two sets gives the resulting set larger number of elements, thus won't affect its $\sigma$-algebra status.
Please help and thank you for your time and effort.
Best Answer
take $X := \{a,b,c\}$ and $A_1 := \{ \{a\}, \{b,c\}, \emptyset, X\}$, $A_2 := \{ \{b\}, \{a,c\}, \emptyset, X\}$ and show that $A_1 \cup A_2$ is not a $\sigma$-algebra