User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)

Set of Countable Ordinals

Transfinite Induction/Schema 1

Proposition 1.23
Let $\mathcal E \subseteq \mathcal P(E)$ be a set of subsets of $E$.

Then the $\sigma$-algebra generated by $\mathcal E$ can be constructed inductively.

The construction is as follows:


 * $\mathcal E_0 = \mathcal E$


 * $\mathcal E_1 = \mathcal E_0 \cup \left({\mathcal P(E) \setminus \mathcal E_0}\right)$

For $j \ge 2, j \in \N$:


 * $\mathcal E_j = \left \{ { \mathcal S \in \mathcal P\left({\mathcal E_{j-1} }\right) : \mathcal S \text { or } \mathcal S^\complement \text{ is countable} }\right\}$

that is, the $\sigma$-algebra of countable and co-countable subsets of sets in $\mathcal E_{j-1}$.


 * $\mathcal E_{\omega} = \displaystyle \bigcup_{j \mathop \in \N} \mathcal E_j$

Let $\Omega$ denote the set of countable ordinals.

Let $\alpha, \beta$ be initial segments in $\Omega$.

Continue the above process by:


 * $\mathcal E_\alpha = \begin{cases} \left \{ { \mathcal S \in \mathcal P\left({\mathcal E_\beta}\right) : \mathcal S \text { or } \mathcal S^\complement \text{ is countable} }\right\} & \alpha \text{ has an immediate predecessor } \beta \\ \bigcup_{\beta \mathop \prec \alpha } \mathcal E_\beta & \text{ otherwise } \end{cases}$


 * $\mathcal E_{\Omega} = \displaystyle \bigcup_{\alpha \mathop \in \Omega} \mathcal E_{\alpha}$

Then $\sigma\left({\mathcal E}\right) = \mathcal E_{\Omega}$.

Question: The structure for finite indices is not strictly necessary because it is subsumed in the construction over cardinals, because natural numbers are finite cardinals. Folland brings it for intuition, not as part of the theorem and proof. Should I leave it as part of the exposition of the theorem? If not, where to put it? --GFauxPas (talk) 12:40, 3 June 2018 (EDT)

Proof
Proposition $1.23$.

Eventually
User:GFauxPas/Sandbox/Zeta2/lnxln1-x/existence

User:GFauxPas/Sandbox/Zeta2/lnxln1-x/integrand

User:GFauxPas/Sandbox/Zeta2/lnxln1-x/evaluation

User:GFauxPas/Sandbox/Zeta2/FourierSeries/

User:GFauxPas/Sandbox/Zeta2/Informal Proof

User:GFauxPas/Sandbox/NumberTheory