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.

Pointwise Operations on Mappings
Shortly, I will need a lot of pointwise operations on mappings, so much that I feel there is a case for a page 'Pointwise Operations on Mappings' or something similar. A quick pick (for (extended) real-valued fns):
 * $f+g$, $f g$, $\max(f,g)$, $\sup_i f_i$, $\limsup_i f_i$, $\lim_{i\to\infty}f_i$

and also notions like pointwise limit of mappings $\lim_j f_j = f$

It feels crafted and is unpleasant to keep introducing these every time they are used.

Maybe I could write pages for each of them (specifically for (extended) real-valued functions), and bundle these on 'pointwise operations on real-valued mappings' or something similar. I hesitate a bit because with my natural ability for abstraction, I don't need these myself; they are adequately formulated on Operation Induced on Set of Mappings for anyone capable of the required abstraction. It's just that a (relatively) self-contained treatment (of eg. analysis or measure theory) should not need to delve into the intricacies of abstract algebra when such isn't necessary.

More pages of this sort could be created, and put on ref'd abstract algebra page as examples. --Lord_Farin 07:51, 5 April 2012 (EDT)

Naming problems
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)


 * Just thought that it would be best to introduce this on Set Definition by Predicate (which, on a side note, needs to be brought up to our evolved standards). Any comments? --Lord_Farin 07:56, 5 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)