Difference between revisions of "Pre-measure/New page"
From Maths
(My work isn't worth saving on the pre-measure page, so doing it here instead.) |
(Completely redone page. Saving work, still to do: conventions and terminology) |
||
| Line 1: | Line 1: | ||
| − | {{ | + | {{Requires references|grade=C|msg=I've sort of forged my own path with regards to what we call a pre-measure. This is because I have found it easier to consider a pre-measure as a very similar thing to a [[measure]], and then shown what other authors call pre-measures extend uniquely to my kind of pre-measure. They also call my kind of pre-measure a pre-measure.}} |
| − | + | ||
==Definition== | ==Definition== | ||
| − | + | <div style="display:none;">{{Extra Maths}}</div>Given a [[ring of sets]], {{M|\mathcal{R} }}, a ''(non-negative) pre-measure'' is an ''[[extended real valued]]'' [[countably additive set function]], {{M|\bar{\mu}:\mathcal{R}\rightarrow\overline{\mathbb{R}_{\ge 0} } }} with the following properties: | |
| − | + | # {{M|1=\bar{\mu}(\emptyset)=0}} | |
| + | # {{M|1=\forall(A_n)_{n=1}^\infty\subseteq\mathcal{R} }}[[pairwise disjoint|{{M|\text{ pairwise disjoint} }}]]{{MM|1=\left[\left(\bigudot_{n=1}^\infty A_n\right)\in\mathcal{R}\implies\left(\bar{\mu}\left(\bigudot_{n=1}^\infty A_n\right)=\sum_{n=1}^\infty\bar{\mu}(A_n)\right)\right]}}<ref group="Note">There is a slight abuse of notation here, by the nature of [[implies]] if the LHS is false, we do not care if the RHS is true or false! However if the LHS is false here, the RHS doesn't even make sense (as {{M|\bar{\mu}(A)}} makes no sense when {{M|A}} is not in the domain of {{M|\bar{\mu} }})</ref><ref group="Note">By using {{M|\bigudot}} we are making it clear that the sequence is that of pairwise disjoint sets and going forward we shall not write "pairwise disjoint" when the [[union]] symbol implies it. For clarity the full statement is thus: | ||
| + | * {{MM|1=\forall(A_n)_{n=1}^\infty\subseteq\mathcal{R}\left[\underbrace{\left(\forall i,j\in\mathbb{N}[i\ne j\implies A_i\cap A_j=\emptyset]\right)}_{\text{the }S_n\text{ are pairwise disjoint} }\overbrace{\wedge}^\text{and}\left(\left(\bigudot_{n=1}^\infty A_n\right)\in\mathcal{R}\implies\left(\bar{\mu}\left(\bigudot_{n=1}^\infty A_n\right)=\sum_{n=1}^\infty\bar{\mu}(A_n)\right)\right)\right]}}</ref> | ||
| + | {{Caution|Be aware that this convention differs from what many authors define as a pre-measure (although it meets their definition) see [[#Conventions and terminology|conventions and terminology]] below}} | ||
| + | ==Conventions and terminology== | ||
| + | |||
==See also== | ==See also== | ||
| − | * [[Measure]] | + | * [[Pre-measurable space]] |
| − | * [[ | + | * [[Measure]] - which we obtain by: |
| − | * [[ | + | ** [[Extending pre-measures to outer-measures]] as a part of |
| + | ** [[Extending pre-measures to measures]] | ||
| + | * [[Semi-ring of sets]] | ||
| + | ** [[Pre-measure on a semi-ring]] | ||
| + | * [[Types of set algebras]] | ||
| + | |||
| + | ==Notes== | ||
| + | <references group="Note"/> | ||
==References== | ==References== | ||
<references/> | <references/> | ||
Latest revision as of 21:25, 17 August 2016
Grade: C
This page requires references, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable, it just means that the author of the page doesn't have a book to hand, or remember the book to find it, which would have been a suitable reference.
The message provided is:
The message provided is:
I've sort of forged my own path with regards to what we call a pre-measure. This is because I have found it easier to consider a pre-measure as a very similar thing to a measure, and then shown what other authors call pre-measures extend uniquely to my kind of pre-measure. They also call my kind of pre-measure a pre-measure.
Definition
Given a ring of sets, [ilmath]\mathcal{R} [/ilmath], a (non-negative) pre-measure is an extended real valued countably additive set function, [ilmath]\bar{\mu}:\mathcal{R}\rightarrow\overline{\mathbb{R}_{\ge 0} } [/ilmath] with the following properties:- [ilmath]\bar{\mu}(\emptyset)=0[/ilmath]
- [ilmath]\forall(A_n)_{n=1}^\infty\subseteq\mathcal{R}[/ilmath][ilmath]\text{ pairwise disjoint} [/ilmath][math]\left[\left(\bigudot_{n=1}^\infty A_n\right)\in\mathcal{R}\implies\left(\bar{\mu}\left(\bigudot_{n=1}^\infty A_n\right)=\sum_{n=1}^\infty\bar{\mu}(A_n)\right)\right][/math][Note 1][Note 2]
Caution:Be aware that this convention differs from what many authors define as a pre-measure (although it meets their definition) see conventions and terminology below
Conventions and terminology
See also
- Pre-measurable space
- Measure - which we obtain by:
- Semi-ring of sets
- Types of set algebras
Notes
- ↑ There is a slight abuse of notation here, by the nature of implies if the LHS is false, we do not care if the RHS is true or false! However if the LHS is false here, the RHS doesn't even make sense (as [ilmath]\bar{\mu}(A)[/ilmath] makes no sense when [ilmath]A[/ilmath] is not in the domain of [ilmath]\bar{\mu} [/ilmath])
- ↑ By using [ilmath]\bigudot[/ilmath] we are making it clear that the sequence is that of pairwise disjoint sets and going forward we shall not write "pairwise disjoint" when the union symbol implies it. For clarity the full statement is thus:
- [math]\forall(A_n)_{n=1}^\infty\subseteq\mathcal{R}\left[\underbrace{\left(\forall i,j\in\mathbb{N}[i\ne j\implies A_i\cap A_j=\emptyset]\right)}_{\text{the }S_n\text{ are pairwise disjoint} }\overbrace{\wedge}^\text{and}\left(\left(\bigudot_{n=1}^\infty A_n\right)\in\mathcal{R}\implies\left(\bar{\mu}\left(\bigudot_{n=1}^\infty A_n\right)=\sum_{n=1}^\infty\bar{\mu}(A_n)\right)\right)\right][/math]
References
| ||||||||||||||||||||
