User:Lord Farin/Sandbox

This page exists for me to be able to test out features I am developing. Also, incomplete proofs may appear here.

Feel free to comment.

Over time, stuff may move to User:Lord_Farin/Sandbox/Archive.

Naming problems
I have been thinking about how to name pages proving (under suitable circumstances) that: where $f$ denotes a Stieltjes function and $\mu$ the associated measure.
 * $f_{\mu_f} = f$
 * $\mu_{f_\mu} = \mu$

However, I can't come up with suitable, reasonably short names. Suggestions will be appreciated. --Lord_Farin 05:05, 3 April 2012 (EDT)


 * Another one: Schilling introduces the following very convenient shorthands:
 * $\{u \le v\} := \{x \in \R: u (x) \le v (x)\}$
 * and similar, about anything you can think of ($\{u = \lambda\}:= \{x\in\R: u(x)=\lambda\}$ and so on). But how to incorporate this into PW? It is really very suitable, eg when writing $\int_{\{u\le v\}}d\mu$ and $\chi_{\{u\le v\}}$, paramount in measure theory. --Lord_Farin 17:33, 3 April 2012 (EDT)

Carathéodory's Theorem
For every $A \subseteq X$, denote with $\mathcal C \left({A}\right)$ the collection of countable $\mathcal S$-covers $\left({S_n}\right)_{n \in \N}$.

Next, define the mapping $\mu^*: \mathcal P \left({X}\right) \to \overline{\R}$ by:


 * $\forall A \subseteq X: \mu^* \left({A}\right) = \inf \ \left\{{\displaystyle \sum_{n \in \N} \mu \left({S_n}\right): \left({S_n}\right)_{n \in \N} \in \mathcal C \left({A}\right) }\right\}$

Here, it is understood that the infimum is taken in the extended real numbers.

Hence, by Infimum of Empty Set, $\inf \varnothing = +\infty$.

Lemma 1
$\mu^*: \mathcal P \left({X}\right) \to \overline{\R}$ is an outer measure.

Lemma 2
For all $S \in \mathcal S$, have $\mu^* \left({S}\right) = \mu \left({S}\right)$

Proof
Next, define a collection $\mathcal A^*$ of subsets of $X$ by:


 * $\mathcal A^* := \left\{{A \subseteq X: \forall B \subseteq X: \mu^* \left({B}\right) = \mu^* \left({B \cap A}\right) + \mu^* \left({B \setminus A}\right)}\right\}$

Let $S,T \in \mathcal S$. Then reason as follows:

Would the result follow from Induced Outer Measure is Outer Measure, Induced Outer Measure Restricted to Semiring is Pre-Measure, Elements of Semiring are Measurable with Respect to Induced Outer Measure, and Outer Measure Restricted to Measurable Sets is Measure (once the pages are finished)? Are those the lemmas needed? –Abcxyz (talk | contribs) 14:47, 23 March 2012 (EDT)
 * Yes, and that Measurable Sets of Outer Measure form Sigma-Algebra or whatever it is called, which is already up. --Lord_Farin 18:43, 23 March 2012 (EDT)
 * It is Measurable Sets are a Sigma-Algebra of Sets. Should the page name be changed to what you (Lord Farin) wrote to have the explicit reference to an outer measure? –Abcxyz (talk | contribs) 19:16, 23 March 2012 (EDT)
 * Not at this point. That may be justified once the foundations and refactorisations are in place and we can get to properly naming pages. As of now, it would only at best be replacing the one idiosyncrasy with the other. Good job breaking this proof into multiple stages, each with merit for their own page. Do you mind posting them, too? --Lord_Farin 19:36, 23 March 2012 (EDT)
 * I wouldn't mind posting them. (I believe I have the proofs, unless I messed up somewhere.) Of course, I wouldn't mind anybody else posting them either. –Abcxyz (talk | contribs) 20:18, 23 March 2012 (EDT)
 * We're also going to have to include the part with uniqueness. By the way, I won't edit ProofWiki tomorrow because I'll be out of town. –Abcxyz (talk | contribs) 20:31, 23 March 2012 (EDT)

Uniqueness is just an application of Uniqueness of Measures; no problem there. --Lord_Farin 03:26, 24 March 2012 (EDT)

Generated Sigma-Algebras
Let $X$ be a set, and let $\mathcal G \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$.

Let $A \subseteq X$ be a subset of $X$.

Then we have the following equality of $\sigma$-algebras on $A$:


 * $\sigma \left({\mathcal G}\right)_A = \sigma \left({A \cap \mathcal G}\right)$

where $\mathcal{A}_A$ denotes the trace $\sigma$-algebra, and $\sigma \left({\mathcal G}\right)$ denotes the $\sigma$-algebra generated by $\mathcal G$.

Comment
I can prove this, but I need a rather technical result (of which I have a reference) that generated sigma-algebras can be obtained by transfinite induction to be able to apply distributivity of intersection. I would rather like to use more elementary means and save the characterisation of generated sigma-algebras for a later moment. Does anyone have an idea (one inclusion is trivial)? --Lord_Farin 06:56, 15 March 2012 (EDT)