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I am working on a problem like this:

Suppose ##\mathscr A_1 \subset \mathscr A_2 \subset \ldots## are sigma-algebras consisting of subsets of a set ##X##. Give an example that ##\bigcup_{i=1}^{\infty} \mathscr A_i## is not sigma-algebra.

I was told to work along finite sigma-algebras on ##\mathbb N##, such that their union contains all singletons. For example, let ##F_n := \{\{1\},\{2\},…,\{n\}\}##, and then let ## \sigma (F_n)## be the sigma-algebra.

Here are what I got so far, making it as simple as possible to ##n = 2## only

(1) Let ##F_1 := \{1\}##, then ##\sigma(F_1) = \{ \emptyset, \{ 1\}, \{ 1\}^c, \mathbb N \}##

(2) Let ##F_2 := \{\{1\}, \{2\}\}##, then ##\sigma(F_2) = \{ \emptyset, \{ 1\}, \{ 1\}^c, \{ 2\}, \{ 2\}^c\ \{ 1, 2\}, \{ 1, 2\}^c ,\mathbb N \}##

(3) Here ##\sigma(F_1) \subset \sigma(F_2)##

(4) But here ##\sigma(F_1) \cup \sigma(F_2) = \sigma(F_2)##, which is a sigma-algebra. Therefore it looks like my example fails.

What was wrong with my analysis? If the above was hopelessly wrong, could you please give me another example? Thank you for your time and help.

Suppose ##\mathscr A_1 \subset \mathscr A_2 \subset \ldots## are sigma-algebras consisting of subsets of a set ##X##. Give an example that ##\bigcup_{i=1}^{\infty} \mathscr A_i## is not sigma-algebra.

I was told to work along finite sigma-algebras on ##\mathbb N##, such that their union contains all singletons. For example, let ##F_n := \{\{1\},\{2\},…,\{n\}\}##, and then let ## \sigma (F_n)## be the sigma-algebra.

Here are what I got so far, making it as simple as possible to ##n = 2## only

(1) Let ##F_1 := \{1\}##, then ##\sigma(F_1) = \{ \emptyset, \{ 1\}, \{ 1\}^c, \mathbb N \}##

(2) Let ##F_2 := \{\{1\}, \{2\}\}##, then ##\sigma(F_2) = \{ \emptyset, \{ 1\}, \{ 1\}^c, \{ 2\}, \{ 2\}^c\ \{ 1, 2\}, \{ 1, 2\}^c ,\mathbb N \}##

(3) Here ##\sigma(F_1) \subset \sigma(F_2)##

(4) But here ##\sigma(F_1) \cup \sigma(F_2) = \sigma(F_2)##, which is a sigma-algebra. Therefore it looks like my example fails.

What was wrong with my analysis? If the above was hopelessly wrong, could you please give me another example? Thank you for your time and help.

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