Math 2.0

Real functions Riemann and related integrals Theorem

Lebesgue's criterion for Riemann integrability

Builds on Riemann integrable functionContinuity at a point

Statement

Let a<ba < b and let f ⁣:[a,b]→ℝf \colon [a, b] \to \mathbb{R} be bounded. Then ff is Riemann integrable if and only if the set of points where ff is not continuous has Lebesgue measure zero.

Remarks

A set E⊆ℝE \subseteq \mathbb{R} has measure zero if for each ε>0\varepsilon > 0 it is covered by countably many open intervals of total length less than ε\varepsilon. Countable sets have measure zero. So Thomae's function is integrable, and so is each monotone function. The Dirichlet function is discontinuous everywhere, so it is not integrable. Giuseppe Vitali and Henri Lebesgue proved the criterion independently, in 1907.

Reading path

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionPartition of an interval, definition, no evidence levelPartition of an inter…definitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionMesh of a partition, definition, no evidence levelMesh of a partitiondefinitionRiemann sum, definition, no evidence levelRiemann sumdefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionRiemann integrable function, definition, no evidence levelRiemann integrable fu…definitionLebesgue's criterion for Riemann integrability, statement, no evidence levelLebesgue's criterion …statement
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.
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  1. Limit point of a set definition
  2. Partition of an interval definition
  3. Limit of a function definition
  4. Mesh of a partition definition
  5. Riemann sum definition
  6. Continuity at a point definition
  7. Riemann integrable function definition
  8. Lebesgue's criterion for Riemann integrability statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Riemann_integral, Integrability
About this page
Tag0565EGK6XP
URIm2://0565EGK6XP
Aliaseslebesgue-vitali-theorem
Kindstatement, theorem
WritingW0 stub
MSC26A42
StatusThis copy of the site has no build data. Run m2 status 0565EGK6XP.
UsesRiemann integrable function, Continuity at a point

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