# What is the meaning of countable sequence of sets?

I am reading the below definitions:

The item S3. says, $$\{A_i\}_{i\geq 1}$$ is any countable sequence of sets. I don't know what the author is trying to say. Does he mean that each $$A_i$$ is countable set or is he saying that the set $$\{A_1, A_2, A_3,...,A_{\infty}\}$$ is countable?

• It is just a sequence. Indexed by positive integers. The sentence would be just as good without the word "countable" unless the book considers "sequences" indexed by more complicated creatures. The point was maybe to emphasize that there can be infinitely many. Commented Aug 26, 2022 at 10:26
• No, I'm not saying that, because the thing you wrote does not exist. There is no such thing as $A_\infty$. If anything, I might be saying that $\{A_1,A_2,A_3,\ldots\}$ is a countable set, but that is kind of a void statement since it is by definition. It is equivalent to the fact that the index set is countable, and that is the point. Commented Aug 26, 2022 at 10:30
• "The union of countably many sets from F is again in F" is the best way to phrase it. Commented Aug 26, 2022 at 10:33
• Yes, the family $\{ A_i \}_{ i \ge 1}$ has countable many elements; no assumption about the number of elements of each set $A_i$. We call it sequence because the family is indexed by the set of natural numbers, that as an order; in case of a generic index set $I$, the family $\{ A_i \}_{i \in I}$ is not a "sequence". Commented Aug 26, 2022 at 13:05
• @DrimitiveWatson For instance, consider family ${\left( a_i \right)}_{i \in \mathbb{R}}$ defined by $a_i = \lfloor i \rfloor$, where $\lfloor \rfloor$ means "floor" or "integer part". Then, the set $\{ a_i ,\, i \in \mathbb{R} \}$ is countable, but the family ${\left( a_i \right)}_{i \in \mathbb{R}}$ is uncountable, and because it is uncountable, we do not call it a sequence.
– Stef
Commented Aug 26, 2022 at 16:29

By definition, a set is countable if its cardinality is the same as the cardinality of the set of natural numbers, $$\mathbb{N}=\{1,2,...,n,...\}$$. The cardinalities of two sets are equal if there is a bijection between them. Hence, if we have a bijection between a set $$A$$ and $$\mathbb{N}$$, then we can "count" elements of $$A$$ using natural numbers.

In this case, the notation $$\{A_i\}_{i \geq 1}$$ (I prefer to write it as $$(A_i)_{i \geq 1}$$ to emphasize that order matters) is just a shortcut for an ''infinite'' tuple $$A = (A_1, A_2,..., A_n, \ldots )$$ which is countable by definition : we have a bijection $$f : A \mapsto \mathbb{N} : \forall i : A_i \in A,~ \ f(A_i)=i \in \mathbb{N}$$.

P.S.

Note that most authors (at least in the books about Real Analysis) define a $$K$$-valued sequence as a function from $$\mathbb{N}$$ to a set $$K$$ (in the context of Real Analysis, often $$K = \mathbb{R}$$ or $$\mathbb{C}$$, in the context of the topic, $$K = \mathcal{F}$$). So to be more formally, to define a sequence $$f : \mathbb{N} \mapsto K$$ one should write that $$\{A_i\}_{i \geq 1} \triangleq (A_i)_{i \geq 1} \triangleq (a_1, a_2,..., a_n, \ldots ) \triangleq \{(n, a_n) \mid n \in \mathbb{N}\}$$ where $$a_n \triangleq f(n) \in K.$$

P.S.S.

In the previous version of this post I wrote that "$$\{A_i\}_{i \geq 1}$$ is just a short notation for a set $$\{A_1, A_2, \ldots, A_n, \ldots\}$$", but it was a mistake. As @Stef noted in the comments below the original post, a sequence is not a set. A simple example is a numerical sequence $$(a_n)_{n\geq 0}$$, where for any $$n \in \mathbb{N},$$ $$a_n =1$$. If we write this sequence in the erroneous form, it will be $$\{1,1,1,1,\ldots\}=\{1\}$$ because in a set, no duplicates are allowed. Thank you @KGhatak for bringing my attention to this.

• Denoting a sequence as a set may not be the right thing to do -- after all, the elements of sets are unique whereas this need not to be the case in sequences. Also check the example of $\{ a_i ,\, i \in \mathbb{R} : a_i = \lfloor i \rfloor \}$ provided by Stef. Commented Jun 17 at 10:46
• @KGhatak thank you. I tried to address your notes in my edit of the post. Commented Jun 17 at 15:13
• @KGhatak: Regarding whether countable additivity is defined by the use of sequences (technically, $\omega$-sequences if context is not clear) or countable sets (where countable could mean "countably infinite" or countable could mean "finite or countably infinite"), all versions can be found in the literature, although in some cases an author can be sloppy by conflating one version with another. However, for the purposes of countable additivity, all versions are equivalent -- see Technical Note in this MSE answer. Commented Jun 17 at 17:04