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:

1. Books:Measures, Integrals and Martingales - René L. Schilling
2. Books:Real and Abstract Analysis - Edwin Hewitt & Karl Stromberg
3. Books:Measure Theory - Volume 1 - V. I. Bogachev
4. Books:Measure Theory - Paul R. Halmos
5. 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...

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...)