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I saw the following definition of sigma algebra in Amann/Escher's Analysis iii.

A subset $\mathcal{A}$ of $\mathfrak{P}(X)$ is called a $\sigma$-algebra over X if it satisfies the properties

i) $X \in \mathcal{A}$;

ii) $A \in \mathcal{A} \Rightarrow A^c \in \mathcal{A}$;

iii) $(A_j) \in \mathcal{A}^\mathbb{N} \Rightarrow \bigcup_{j \in \mathbb{N}} A_j \in \mathcal{A}$.

The first two properties is rather conventional. But I am confused with the notation of the third property since it's very different from those I have seen from other books. I think it's trying to convey the idea that a countable union of elements in sigma algebra is also an element in sigma algebra.

My understanding here is that if $(A_j) \in \mathcal{A}^\mathbb{N}$ then $A_j$ is a N-tuple whose elements are subsets of X. How come their countable union is a member of $\mathcal{A}$? Could some please explain how does it work to me? Thanks very much in advance.

Mirabelle
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You've basically got it already! $(A_j)$ is not an $N$-tuple for some finite number $N$, but rather an "$\Bbb N$-tuple"—that is, a sequence—of subsets of $X$. And yes, $\bigcup_{j\in\Bbb N} A_j$ denotes the union of the countably many sets $A_j$—in other words, it denotes $A_1\cup A_2\cup A_3\cup\cdots$. And that countable union is an element of $\mathcal A$ by the definition of a $\sigma$-algebra—it's not a deduction from something else.

Greg Martin
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  • Thank you very much for your answer Greg! But I think I might still need some clarification here. I get the sequence part now. So what you mean is that the notation $(A_j) \in A^\mathbb{N}$ here actually denotes $A_j$ is a term in the sequence of subsets of X? That makes sense but I have never seen this notation before, is this common? – Mirabelle Apr 08 '22 at 17:20
  • @Mirabelle: The notation is fairly common (see Meaning of a set in the exponent and here), but this probably should have been defined (but maybe is defined earlier in the book?) since it's use in this context is unusual. In fact, to me it's use in this context seems a bit excessively trendy, as if the author was trying to impress the reader. – Dave L. Renfro Apr 08 '22 at 18:00
  • @DaveL.Renfro Thank you for your answer Dave. Yes, I have the same feeling. I do like this book as it's really comprehensive and all. However, the notation has been constantly causing trouble. It's often done in very indirectly manner and different from those in other books. – Mirabelle Apr 08 '22 at 19:12