Math 2.0

Measure and integration Construction of measures Problem

Measure problem of Lebesgue

Builds on Lebesgue measureCountable additivityTranslation

Problem

Is there a function mm, defined on all bounded subsets of ℝ\R, with values in [0,∞)[0, \infty), such that:

  1. mm is countably additive: m(⋃kAk)=∑km(Ak)m\big(\bigcup_k A_k\big) = \sum_k m(A_k) for pairwise disjoint bounded AkA_k whose union is bounded.
  2. mm is invariant under congruence: m(A)=m(B)m(A) = m(B) when BB is a translation or reflection of AA.
  3. m([0,1])=1m([0, 1]) = 1.

Remarks

Lebesgue measure has all three properties on the Lebesgue measurable sets. The answer is negative: Vitali showed in 1905, with the Vitali set and the axiom of choice, that no such mm exists. Banach and Ulam studied the question without congruence invariance. Is there a countably additive measure on all subsets of ℝ\R that is 00 on points and 11 on [0,1][0, 1]? Ulam showed that a positive answer implies that some cardinal at most the continuum is real-valued measurable, a notion close to that of a measurable cardinal. Such a cardinal is weakly inaccessible, so ZFC cannot prove a positive answer, if ZFC is consistent. Dropping countable additivity and keeping finite additivity leads to the Banach measure problem.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Borel sigma-algebra, concept, no evidence levelBorel sigma-algebraconceptCountable additivity, concept, no evidence levelCountable additivityconceptEuclidean space, object, no evidence levelEuclidean spaceobjectOuter measure, concept, no evidence levelOuter measureconceptTranslation, concept, no evidence levelTranslationconceptLebesgue measure, object, no evidence levelLebesgue measureobjectMeasure problem of Lebesgue, problem, no evidence levelMeasure problem of Le…problem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Borel sigma-algebra concept
  2. Countable additivity concept
  3. Euclidean space object
  4. Outer measure concept
  5. Translation concept
  6. Lebesgue measure object
  7. Measure problem of Lebesgue problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Measurable_cardinal
  2. Web https://en.wikipedia.org/wiki/Non-measurable_set
About this page
TagTY12G0E8KX
URIm2://TY12G0E8KX
Kindproblem
WritingW0 stub
MSC28A12, 28E15, 03E55
StatusThis copy of the site has no build data. Run m2 status TY12G0E8KX.
AboutLebesgue measure
UsesLebesgue measure, Countable additivity, Translation
Changed2 November 2026

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