View source for Notes:Measures

==Problem==
It seems no one can agree on quite ''what'' a measure is. This page is intended to be a gathering of opinions from a few authors to see what is what. Bogachev for example (author of [[Books:Measure Theory - Volume 1 - V. I. Bogachev]]) doesn't require that a measure even be positive! 
Books used:
# [[Books:Measures, Integrals and Martingales - René L. Schilling]]
# [[Books:Real and Abstract Analysis - Edwin Hewitt & Karl Stromberg]]
# [[Books:Measure Theory - Volume 1 - V. I. Bogachev]]
# [[Books:Measure Theory - Paul R. Halmos]]
# [[Books:Analysis - Part 2: Integration, Distributions, Holomorphic Functions, Tensor and Harmonic Analysis - Krzysztof Maurin]]

==Definitions==
===Bogachev===
* ''Measure'' - [[countably additive set function]] defined on an [[algebra of sets]], [[Real valued function|Real valued]]
No notion of [[pre-measure]]
===Measures, Integrals and Martingales===
* ''Measure'' (positive) is an [[extended real valued function|extended real valued]] positive [[countably additive set function]] on a [[sigma-algebra]] 
* ''Pre-measure'' (positive) is the same thing but on an [[algebra of sets]] (Bogachev's measure, Real and Abstract Analysis)
===Maurin===
Not applicable (not sure what magic he's up to....)
===Halmos===
* ''Measure'' - [[Extended real valued function|extended real valued]] non-negative [[countably additive set function]] defined on a [[ring of sets]]
===Real and Abstract Analysis===
* ''Outer measure'' - as a thing defined on the powerset of a set with some properties.
* ''Measure'' - [[extended real valued function|extended real valued]] non-negative [[countably additive set function]] defined on an [[algebra of sets]]
* ''Measure space'' - measure on a {{sigma|algebra}}

==Proposal==
TODO!


==Discussion==
As for me, one usually says "measure" for a sigma-additive function with values in {{M|[0,\infty]}} on a sigma-algebra. For other cases, one adds something. Yes, I know, you hate this "non-monotone terminology", and I can understand your feeling, but I doubt we can change the world... I guess, the reason is, a conflict between two kinds of monotonicity in terminology: logical monotonicity (shorter name for more general notion) and pragmatical monotonicity (shorter name for notion that is used more often).

Namely, one says: "measure on algebra"; "signed measure"; "complex measure"; "vector measure"; "finitely additive measure". All these are not really measures. [[User:Boris|Boris]] ([[User talk:Boris|talk]]) 21:19, 20 March 2016 (UTC)

(Is it OK to put here a signed message? This is not a talk page; but this is a note page...)
: It's better than fine, your input is actually great seeing as how this is your area! I'm also inclined to agree. In the ancient article [[Measure Theory]] which I doubt is complete I put forward the idea of "if you want to measure stuff, you want these rules (set subtraction, union)" and [[pre-measure]] is a sensible sounding thing. Measure should denote the thing we actually want to measure on! We make up the terminology remember, there's no "natural" measure out there (I hope you know what I mean by this). 
:* Can I measure the limit of {{M|(0,1-\frac{1}{n})}}, sure
:* Can I measure the limit of {{M|(0,n)}} - sure (we need {{M|+\infty}} for this)
: With this in mind I like:
:* '''Pre-measure''' - positive (extended?) real-valued additive function defined on a [[ring of sets]]<ref group="Note">Every [[algebra of sets]] is also a [[ring of sets]], Bogachev suggests this is okay to do. Also in the [[Measure Theory]] page I mentioned, I never could see why a (sigma) ''algebra'' was needed when a ''ring'' would do.</ref> - this is in line with [[Books:Measures, Integrals and Martingales - René L. Schilling]] which is extremely well written and very formal. 
:* '''Measure''' - positive extended-real-valued countably/sigma-additive function defined on a [[sigma-ring]]
: BUT I need to answer some questions first:
:# Are negative measures a thing? [[Pre-measure/Properties in common with measure|Sub-additive (mentioned here)]] is like a [[triangle inequality]] - which if we want requires the measure to be positive.
:# The outer-measure seems to be defined on the powerset of the space. Is this the same as the sigma-algebra on the reals usually? If I have a pre-measure (on the ring of all half-open-half-closed rectangles) how do I go about extending that to a measure? 
:# Do you see any problems with this?
: I'm now going to go through Halmos' Measure Theory, then Measures Integrals and Martingale's relevant sections to decide. 
: [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 23:47, 20 March 2016 (UTC)

==Notes (yes on a notes page)==
<references group="Note"/>

==See also==
* [[Notes:Things we want to measure on]]

[[Category:Measure Theory]]