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Research draft

measure

vr.tr.measure · XCT.QLT

Let an agent handle measures by mathematical definition, examples and applications, and separate the everyday senses of measure.

Thing Registry Cross-cutting context

Research draft, second pass

A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.

written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify

Researched by: Claude

Purpose and description

Let an agent handle measures by mathematical definition, examples and applications, and separate the everyday senses of measure.

In mathematics, a function that assigns a non-negative size, such as length, area, volume or probability, to subsets of a space in a countably additive way, as in Lebesgue measure; the term is also used loosely for any quantifying function or standard of measurement.

What it is for: Integration, probability and quantifying size.

It can be state the definition of a measure; give examples such as Lebesgue and counting measures; relate measures to integration and probability; separate everyday senses.

Distinguishing features

Non-negative

Countably additive

Defined on a sigma-algebra

Basis of modern integration

What it looks like

Not physical; written as functions on sets.

How it is recognised

Notation mu(A)

Sigma-algebras and measurable sets

A measuring instrument or legislative measure are other senses

Related models

is a kind of - category

function

is studied in - field

measure theory

underpins - application

probability

is used in - application

Lebesgue integration

In practice

Families and kinds

Lebesgue measure

counting measure

probability measures

Borel measures

Hausdorff measure

Standards and regulation

ISO 80000-2 notation

Failure modes and hazards

Assuming every set is measurable

Confusing senses

Also called

calibration standardpyrometric devicemass per unit lengthgeometric measurecompactness measure of a shaperoundnessaccuracy and precisionin essenceliterallyunsharpnessposition accuracyF1 scoresigned measureflow measurementcomplex measureHaar measureDirac measureimportanceregular measureproduct measureHausdorff measuredecomposable measureJordan measureSingular measureLebesgue–Stieltjes measuremultiple of the medianabsolutely continuous measurelocally finite measurevector measuretrivial measurerandom measuremeasure of associationFlow cupscounting measureGibbs measureBesov measureSub-probability measureBaire measureouter regular measureIsotropic measure

+44

Where this came from

wikidata · CC0 1.0

Source repository Registry entry point

Also registered as vr.tr.measure-artifact

Drafted structure

Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.

Definition Axioms.

Axioms define measures.

Axioms

Conditions.

Axioms

Axioms.

  1. Does this function satisfy the measure axioms? boundary
  2. On which sigma-algebra is it defined? definition

Examples

Common measures.

Examples

Examples.

  1. Which standard measure fits this problem? definition
  2. How is it constructed? definition
Theory Results.

Theory gives tools.

Integration

Lebesgue integral.

Integration

Integration.

  1. How is the integral defined with respect to this measure? definition
  2. Which convergence theorem applies? provenance

Non-measurable

Limits.

Non-measurable

Non-measurable sets.

  1. Could the set be non-measurable? boundary
  2. What does that imply? definition
Applications Uses.

Measures are everywhere.

Probability

Probability measures.

Probability

Probability measures.

  1. How does a probability measure formalise this problem? definition
  2. What is the sample space? definition

Geometry

Size of sets.

Geometry

Geometric measure.

  1. What is the measure of this set, such as its area or dimension? measurement
  2. Which measure is used? definition
Senses Disambiguation.

Measure has many senses.

Everyday

Measuring and units.

Everyday

Everyday sense.

  1. Is a unit, instrument or act of measuring meant? boundary
  2. Which entry fits? action

Other

Music and law.

Other

Other senses.

  1. Is a musical bar or a legislative measure meant? boundary
  2. Which entry fits? action

What the second pass must settle

  • Should specific measures be separate entries?
  • How should measure theory be linked?
  • How should everyday senses be split off?
Draft as JSON