Oddly enough, I didn't find existing question/answer about this. I want to know the difference because that's what makes $\sigma$-field different from field (i.e., the former is closed under countable union whereas the latter is closed under union). Why union doesn't imply countable union through induction?
Asked
Active
Viewed 40 times
0
-
2See https://math.stackexchange.com/questions/150530/sigma-algebra-and-algebra-difference for example. – Ats Jan 12 '23 at 15:24
-
Ah I didn't know I should have searched 'algebra' instead of 'field'. thanks – Sam Jan 12 '23 at 17:15
-
If $\mathcal A$ is closed under (binary) unions and $A_n\in\mathcal A$ for $n=1,2,\dots$ then with induction it can be proved that $\bigcup_{k=1}^nA_k\in\mathcal A$ for every positive integer $n$. However it cannot be proved by induction that $\bigcup_{k=1}^{\infty}A_k\in\mathcal A$. – drhab Jan 12 '23 at 21:58