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)

For Equivalence of Definitions of Sigma-Algebra

Definition 2 implies Definition 1
Let $\mathcal R$ be a system of sets on a set $X$ such that:
 * $(1): \quad X \in \mathcal R$
 * $(2): \quad \forall A, B \in \mathcal R: \complement_X \left({A}\right) \in \mathcal R$
 * $(3): \quad \displaystyle \forall A_n \in \mathcal R: n = 1, 2, \ldots: \bigsqcup_{n \mathop = 1}^\infty A_n \in \mathcal R$

Conditions $(1)$ and $(2)$ in definition 2 are identical to that of conditions $(1)$ and $(2)$ in definition 1.

For condition $(3)$, let $\left\{ { E_j} \right \}_{j \mathop = 0}^\infty$ be a countable set of sets in $\mathcal R$ indexed by $\N$.

Define another set $\displaystyle \left\{ { F_k } \right \}_{j \mathop = 0}^\infty$ by:


 * $\displaystyle F_k = E_k \setminus \left({ \bigcup_{j \mathop = 0}^{k \mathop - 1} E_j }\right)$

By De Morgan's:


 * $\displaystyle F_k = \bigcap_{j \mathop = 0}^{k \mathop - 1} \left({E_k \setminus E_j}\right)$

Then note, if $F_m$ and $F_n$ are Definition:Distinct:

something something they're disjoint

a bit stuck on the set arithmetic here. I'll come back to it later, feel free to leave me a hint. --GFauxPas (talk) 13:34, 29 May 2018 (EDT)

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