Difference between revisions of "Doctrine:Measure theory terminology"
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Revision as of 21:12, 20 August 2016
Contents
[hide]Proposals
Splicing sets
I propose that rather than mu*-measurable sets we instead use outer splicing sets or just splicing sets. Currently:
- For an outer-measure, μ∗:H→¯R≥0 we call a set, X∈H, μ∗-measurable if:
- ∀Y∈H[μ∗(Y)=μ∗(Y−X)+μ∗(Y∩X)]
μ∗-measurable must be said with respect to an outer measure (μ∗) and is very close to "outer measurable set" which would just be an set the outer measure assigns a measure to[Note 1]. However if we call X a splicing set then all ambiguity goes away and the name reflects what it does. In a sense:
- X is a set that allows you to "splice" (the measures of) Y−X and Y∩X together in a way which preserves the measure of Y. That is, the sum of the measures of the spliced parts is the measure of Y.
If there is such a thing as μ∗-measurable sets for the inner-measure they can simply be called "inner splicing sets" although I doubt that'll be needed. Alec (talk) 21:14, 20 August 2016 (UTC)
Inner vs outer splicing sets=
I propose that when we speak of just a splicing set it be considered as an outer one (unless the context implies otherwise, for example if only inner-measures are in play) Alec (talk) 21:29, 20 August 2016 (UTC)
Standard symbols
- S∗ for the set of all (outer) splicing sets with respect to the outer-measure μ∗ say, of the context.
- S∗ for the set of all inner splicing sets with respect to the inner-measure μ∗ say, of the context. Caution:Should such a definition make sense.
Points to address
- Is there such a thing as "inner splicing sets"?
- There does not appear to be a corresponding notion for inner-measures however there are similar things (see page 61 of Halmos' measure theory) in play
- Does "splicing set" arise anywhere else?
Notes
- Jump up ↑ Not every set is outer-measurable unless H is the powerset of the "universal set" in question