In what follows \(\displaystyle \bigsqcup\) denotes disjoint union. I have marked the point where I don't understand the reasoning with (WHY).
In the proof([Measures integrals and martingales][1] starts p 39)(step 2): the following set is defined \(\displaystyle \mathcal S_\cup=\{S_1 \sqcup S_2 \ldots \sqcup S_M : M\in \Bbb{N}, S_j \in \mathcal S \}\) where \(\displaystyle \mathcal S\) is a [semi ring][2]. Now let the collection of sets \(\displaystyle (T_k)_{k\in \Bbb{N}} \subset \mathcal S_\cup\) be pairwise disjoint such that \(\displaystyle T:= \bigsqcup_{k \in \Bbb{N}}T_k \in \mathcal S_\cup\). It can be shown that \(\displaystyle \mathcal S_\cup\) is a [ring][3]. (Not sure how this will help)
By the definition of \(\displaystyle \mathcal S_\cup\) we find \(\displaystyle (S_k)_{k\in \Bbb{N}}\subset \mathcal S\) and a sequence of integers \(\displaystyle 0=n(0)\le n(1) \le n(2) \le \dots\) such that \(\displaystyle T_k=S_{n(k-1)+1} \sqcup \ldots \sqcup S_{n(k)}\) and \(\displaystyle T=U_1 \sqcup \ldots \sqcup U_N\) where \(\displaystyle \displaystyle U_l=\bigsqcup_{j\in J_l}S_j \in \mathcal S\)(WHY?) with disjoint index sets \(\displaystyle J_1 \sqcup J_2\ldots \sqcup J_N= \Bbb{N}\) that partition \(\displaystyle \Bbb{N}\).
I was thinking since \(\displaystyle T \in \mathcal S_\cup\) then \(\displaystyle T=S'_1 \sqcup \ldots S'_M\) where \(\displaystyle S'_i \in \mathcal S\) but I can't go anywhere from here(That is I have no idea on how I can prove \(\displaystyle S'_i=U_i\)).
[1]: http://books.google.com/books?id=_O...page&q=measures integrals martingales&f=false
[2]: http://www.proofwiki.org/wiki/Definition:Semiring_of_Sets
[3]: http://en.wikipedia.org/wiki/Ring_of_sets
In the proof([Measures integrals and martingales][1] starts p 39)(step 2): the following set is defined \(\displaystyle \mathcal S_\cup=\{S_1 \sqcup S_2 \ldots \sqcup S_M : M\in \Bbb{N}, S_j \in \mathcal S \}\) where \(\displaystyle \mathcal S\) is a [semi ring][2]. Now let the collection of sets \(\displaystyle (T_k)_{k\in \Bbb{N}} \subset \mathcal S_\cup\) be pairwise disjoint such that \(\displaystyle T:= \bigsqcup_{k \in \Bbb{N}}T_k \in \mathcal S_\cup\). It can be shown that \(\displaystyle \mathcal S_\cup\) is a [ring][3]. (Not sure how this will help)
By the definition of \(\displaystyle \mathcal S_\cup\) we find \(\displaystyle (S_k)_{k\in \Bbb{N}}\subset \mathcal S\) and a sequence of integers \(\displaystyle 0=n(0)\le n(1) \le n(2) \le \dots\) such that \(\displaystyle T_k=S_{n(k-1)+1} \sqcup \ldots \sqcup S_{n(k)}\) and \(\displaystyle T=U_1 \sqcup \ldots \sqcup U_N\) where \(\displaystyle \displaystyle U_l=\bigsqcup_{j\in J_l}S_j \in \mathcal S\)(WHY?) with disjoint index sets \(\displaystyle J_1 \sqcup J_2\ldots \sqcup J_N= \Bbb{N}\) that partition \(\displaystyle \Bbb{N}\).
I was thinking since \(\displaystyle T \in \mathcal S_\cup\) then \(\displaystyle T=S'_1 \sqcup \ldots S'_M\) where \(\displaystyle S'_i \in \mathcal S\) but I can't go anywhere from here(That is I have no idea on how I can prove \(\displaystyle S'_i=U_i\)).
[1]: http://books.google.com/books?id=_O...page&q=measures integrals martingales&f=false
[2]: http://www.proofwiki.org/wiki/Definition:Semiring_of_Sets
[3]: http://en.wikipedia.org/wiki/Ring_of_sets