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.

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:

Sum over Countable Set
It is sometimes useful not having to distinguish between finite and infinite sums, eg. when dealing with a sum over a set which is either finite or countably infinite. I seek for a foundation for the notation $\displaystyle \sum_{\omega\in\Omega} p_\omega$, for example.


 * It appears I have already provided it by posting Definition:Generalized Sum; the stuff works brilliantly.

Extended Reals as Two-Point Compactification
There will be text here explaining this idea, which puts a compactifying topology on $\overline{\R}$, making the notions of diverging to $\pm \infty$ precise, and also allows for more rigorous treatment of, for example, convergence issues in measure theory.

Some search suggest that the required topology is the Definition:Order Topology, which apparently doesn't exist; I will try and write something down. Plainly, it's a topology on a toset generated by all the segments. Whoa, ideas start tumbling in, about $\max$ and $\min$, $\sup$, $\inf$ etc. etc continuous, morphisms in associated categories etc etc. I'd better be satisfied with Characterization of Measures as the milestone for today, or I won't sleep at all tonight. --Lord_Farin 18:59, 16 March 2012 (EDT)


 * Research done, searching for time to post the stuff. --Lord_Farin 14:42, 20 March 2012 (EDT)


 * Will get to it once I have posted the proof of Carathéodory's Theorem (Measure Theory), a long one. --Lord_Farin 11:27, 23 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)