View source for Measure theory terminology doctrine

==Overview==
* [[Pre-measure]], {{M|\bar{\mu} }} - ''[[extended real valued]]'' [[countably additive set function]] defined on a [[ring (measure theory)|ring]].
* [[Outer-measure]], {{M|\mu^*}} - ''[[extended real valued]]'' [[countably subadditive set function|countably '''sub'''additive set function]] defined on a [[hereditary sigma ring|hereditary {{sigma|ring}}]]
* [[Measure]], {{M|\mu}} - ''[[extended real valued]]'' [[countably additive set function]] defined on a [[sigma-ring|{{sigma|ring}}]].

==Methodology==
Inline with the primary measure theory references{{rMTH}}{{rMIAMRLS}}{{rMT1VIB}} there are 3 steps to constructing a [[measure]] (as in the thing that "measures" on a [[sigma-ring|{{sigma|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 [[sigma-ring|{{sigma|ring}}]] and showing that the outer measure restricted to this subset is [[countably additive set function|countably additive]].
Some books work on [[algebra of sets|algebras]] and [[sigma-algebra|{{sigma|algebras}}]] and only mention [[ring of sets|rings]] and [[sigma-ring|{{sigma|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. 

==References==
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[[Category:Measure Theory]][[Category:Doctrine]]