Measure theory terminology doctrine
From Maths
Overview
- Pre-measure, [ilmath]\bar{\mu} [/ilmath] - extended real valued countably additive set function defined on a ring.
- Outer-measure, [ilmath]\mu^*[/ilmath] - extended real valued countably subadditive set function defined on a hereditary [ilmath]\sigma[/ilmath]-ring
- Measure, [ilmath]\mu[/ilmath] - extended real valued countably additive set function defined on a [ilmath]\sigma[/ilmath]-ring.
Methodology
Inline with the primary measure theory references[1][2][3] there are 3 steps to constructing a measure (as in the thing that "measures" on a [ilmath]\sigma[/ilmath]-ring)
- Defining what we call a pre-measure
- Extending it to the outer measure
- Looking at a subset of the outer measurable sets, which happens to be a [ilmath]\sigma[/ilmath]-ring and showing that the outer measure restricted to this subset is countably additive.
Some books work on algebras and [ilmath]\sigma[/ilmath]-algebras and only mention rings and [ilmath]\sigma[/ilmath]-rings, where as others, notably Halmos, deal with rings.
As all algebras are instances of rings and all sigma algebras instances of sigma rings there is no problem in defining all the terminology on rings.
After a discussion with an expert in the field, I shall make it clear that algebras and sigma-algebras MUST be kept on the readers mind.
