I hope to clear up some confusion I have about sigma fields.
For a concrete example, say I list a few arbitrary elements of a sigma field generated by a specific set that is defined on the natural numbers:
$\{1\},\{1,2\},\{1,2,3\}...$
if $A_1=\{1\}$ and $A_2=\{1,2\}$ then $A_1\cup A_2 = \{\{1\}\{1,2\}\}$ correct?
For another example, my confusion with elements of the sigma field is that say on a sigma field generated by intervals on $(0,1]$, I choose the interval $B_1=(a,b)$, then the complement of $B_1$ would be all other intervals in the sigma field?
So the intervals $(0,a), (b,1), (b+c,1),... etc. $ where$(b<c<1)$ would all be included in the complement of $B_1$?
But then also since the sigma field is all useful sets on $(0,1]$ then $(a+k,b+k)$ where $(k<b,b+k<1)$ would also be in the complement of $B_1$. So this interval that is on top of the interval described by $B_1$ but shifted by k is also in the complement of $B_1$?
Then if $$B_1=(a,b)$$ $$B_2=(a+k,b+k)$$
it should follow that,
$$B_1\cap B_2 = \phi$$
Would this be a correct interpretation? Thanks for any help in my understanding!