I read from a book the below example that a countable union of sigma algebra's is not necessarily a sigma algebra.
I have problem understanding why N ∈ F Must mean N ∈ some Fi ? From the construction it seems that N belongs to F because of the limit, not any of the Fi...
Thanks
"Put Ωi = {j}j=1...i, and let Fi be the σ-algebra of the collection of all subsets of Ωi for i ∈ N. Suppose that F = ∪Fi (I=1... ∞ ) is also a σ-algebra. Since, for each i, {i} ∈ Fi and Fi ⊆ F we have {i} ∈ F. Thus, by our primary assumption, N = ∪{i} (I=1... ∞ ) ∈ F and, therefore, N ∈ Fi for some i, which implies N ⊆ Ωi , a contradiction. Hence, F is not a σ-algebra."