Example where union of increasing sigma algebras is not a sigma algebra

Solution 1:

The problem arises in the countable union; your argument is correct as far as it goes, but from the fact that $\cup_{i=1}^n x_i\in \cup_{i=1}^{\infty}F_i$ for each $n$ you cannot conclude that $\cup_{i=1}^{\infty} x_i$ lies in $\cup_{i=1}^{\infty} F_i$: the full union must be in one of the $F_j$ in order to be in $\cup_{i=1}^{\infty}F_i$.

For an explicit example, take $X=\mathbb{N}$; let $F_n$ be the sigma algebra that consists of all subsets of $\{1,\ldots,n\}$ and their complements in $X$. Now let $x_i=\{2i\}$. Then each $x_i$ is in $\cup F_i$, but the union does not lie in any of the $F_k$, hence does not lie in $\cup F_i$.

Added: In this example, $\cup_{i=1}^{\infty}F_n$ is the algebra of subsets of $X$ consisting of all subsets that are either finite or cofinite, so any infinite subset with infinite complement will not lie in the union, and such a set can always be expressed as a countable union of elements of $\cup F_i$.

Solution 2:

Something more drastic is true: If $\langle \mathcal{F}_n: n \geq 1\rangle$ is a strictly increasing sequence of sigma algebras over some set $X$ then $ \bigcup_{n \geq 1} \mathcal{F_n}$ is not a sigma algebra. As a corollary, there is no countably infinite sigma algebra. See, for example, "A comment on unions of sigma fields, A. Broughton, B. Huff, American mathematical monthly, 1977 Vol. 84 No. 7, pp 553-54".

Solution 3:

Let $\Omega=[0,1]$, $A_{0}= \{\emptyset, \Omega \}$ and $A_{k}=\sigma \{[0,\frac{1}{2^k}],[\frac{1}{2^k},\frac{2}{2^k}],[\frac{2}{2^k},\frac{3}{2^k}],.....,[\frac{2^k-1}{2^k},1]\}$

pick irrational number $x\in(0,1)$ and sequence $s_{1},s_{2},...$ converging to $x$ from the left (binary representation allows to find such sequence from $A_{i}$'s). Then $(x,1]=\cap_{i=1}^{\infty}(s_{i},1] \in \cup_{i=0}^{\infty} A_{i}$. Then $\{x\} \in\cup_{i=0}^{\infty} A_{i} $. But for all fixed k, $A_{k}$ contains only rational numbers and intervals.