- #1
tunaaa
- 12
- 0
Hello, first I’d like to clarify that the only difference between an algebra and a sigma-algebra, is that we have
[tex]A,B \in \mathcal{A} \Rightarrow A \cup B \in \mathcal{A} \text{ (1) for } \mathcal{A} \text{ algebra}[/tex][tex]A_1, A_2, A_3, \ldots\in\mathcal{A} \Rightarrow \bigcup_{i=1}^{\infty}A_i \in \mathcal{A}\text{ (2) for } \mathcal{A} \text{ sigma-algebra}[/tex]If this understanding is correct, then what I am confused about is that surely (1) implies (2), since, starting with (1),
[tex]\text{if } A_1, A_2 \in\mathcal{A} \text{ then } A_1 \cup A_2 \in\mathcal{A} \text{ by definition}[/tex][tex]\text{now if also } A_3 \in\mathcal{A} \text{ then } \underbrace{(A_1 \cup A_2)}_{A} \cup \underbrace{A_3}_{B} \in\mathcal{A}[/tex][tex]\text{now if also } A_4 \in\mathcal{A} \ldots[/tex]and so do we not arrive at (2)?
Thanks for your help.
[tex]A,B \in \mathcal{A} \Rightarrow A \cup B \in \mathcal{A} \text{ (1) for } \mathcal{A} \text{ algebra}[/tex][tex]A_1, A_2, A_3, \ldots\in\mathcal{A} \Rightarrow \bigcup_{i=1}^{\infty}A_i \in \mathcal{A}\text{ (2) for } \mathcal{A} \text{ sigma-algebra}[/tex]If this understanding is correct, then what I am confused about is that surely (1) implies (2), since, starting with (1),
[tex]\text{if } A_1, A_2 \in\mathcal{A} \text{ then } A_1 \cup A_2 \in\mathcal{A} \text{ by definition}[/tex][tex]\text{now if also } A_3 \in\mathcal{A} \text{ then } \underbrace{(A_1 \cup A_2)}_{A} \cup \underbrace{A_3}_{B} \in\mathcal{A}[/tex][tex]\text{now if also } A_4 \in\mathcal{A} \ldots[/tex]and so do we not arrive at (2)?
Thanks for your help.