Difference between revisions of "Notes:Measures"
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| − | 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... | + | 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) | 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) | ||
Revision as of 21:34, 20 March 2016
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!
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...)
