Notes:Measures
Contents
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
No notion of pre-measure
Measures, Integrals and Martingales
- Measure (positive) is an 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 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 non-negative countably additive set function defined on an algebra of sets
- Measure space - measure on a [ilmath]\sigma[/ilmath]-algebra
Proposal
TODO!
Discussion
As for me, one usually says "measure" for a sigma-additive function with values in [ilmath][0,\infty][/ilmath] 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. 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 very welcome and better than fine, your input is actually greatly valued 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 [ilmath](0,1-\frac{1}{n})[/ilmath], sure
- Can I measure the limit of [ilmath](0,n)[/ilmath] - sure (we need [ilmath]+\infty[/ilmath] for this)
- With this in mind I like:
- Pre-measure - positive (extended?) real-valued additive function defined on a ring of sets^{[Note 1]} - 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? 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.
- Also; in probability I can observe an event not happening, this suggests I might actually want a sigma-algebra. However as all [ilmath]\sigma[/ilmath]-algebras are [ilmath]\sigma[/ilmath]-rings this should cause no problems.
- Alec (talk) 23:47, 20 March 2016 (UTC)
Notes (yes on a notes page)
- ↑ 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.