Math 2.0

Measure and integration Definition

Bochner integral

Builds on Measure spaceStrongly measurable functionIntegral of a non-negative functionBanach space

Definition

Let (X,Σ,μ)(X, \Sigma, \mu) be a measure space and let EE be a Banach space over ℝ\R or ℂ\C. An integrable simple function is a function s=∑i=1n𝟏Aieis = \sum_{i=1}^{n} \mathbf{1}_{A_i} e_i with ei∈Ee_i \in E and pairwise disjoint Ai∈ΣA_i \in \Sigma with μ(Ai)<∞\mu(A_i) < \infty. Its integral is ∫Xs dμ=∑i=1nμ(Ai) ei\int_X s \, d\mu = \sum_{i=1}^{n} \mu(A_i) \, e_i. A function f ⁣:X→Ef \colon X \to E is Bochner integrable if both of the following hold.

  1. The function ff is strongly measurable.
  2. There are integrable simple functions s1,s2,…s_1, s_2, \dots with lim⁡n→∞∫X∥f−sn∥ dμ=0\lim_{n \to \infty} \int_X \norm{f - s_n} \, d\mu = 0.

The integrals in clause 2 are integrals of non-negative functions. The Bochner integral of ff is then ∫Xf dμ=lim⁡n→∞∫Xsn dμ\int_X f \, d\mu = \lim_{n \to \infty} \int_X s_n \, d\mu, a limit in EE.

Remarks

The limit exists, since ∥∫sn−∫sm∥≤∫∥sn−sm∥\norm{\int s_n - \int s_m} \le \int \norm{s_n - s_m}, so the integrals form a Cauchy sequence in EE. It does not depend on the choice of the sns_n.

Example: let X=[0,1]X = [0, 1] with Lebesgue measure and let f ⁣:X→Ef \colon X \to E be continuous. The step functions sns_n equal to f(k/n)f(k/n) on [k/n,(k+1)/n)[k/n, (k+1)/n) for 0≤k<n0 \le k < n and to f(1)f(1) at 11 converge to ff uniformly, so ∫∥f−sn∥→0\int \norm{f - s_n} \to 0. So ff is Bochner integrable, and its integral is the limit of the Riemann sums 1n∑k=0n−1f(k/n)\frac{1}{n} \sum_{k=0}^{n-1} f(k/n).

Non-example: take X=ℝX = \R with Lebesgue measure, E=ℝE = \R and f=1f = 1. It is strongly measurable, but every integrable simple function vanishes outside a set of finite measure, so ∫∥f−s∥=∞\int \norm{f - s} = \infty for each of them.

Bochner's theorem: a strongly measurable function ff is Bochner integrable if and only if ∫X∥f∥ dμ<∞\int_X \norm{f} \, d\mu < \infty. For E=ℝE = \R or E=ℂE = \C the Bochner integral is the Lebesgue integral. For a bounded linear map T ⁣:E→FT \colon E \to F one has T∫f dμ=∫Tf dμT \int f \, d\mu = \int T f \, d\mu.

Reading path

To understand this page from the ground up, read these 40 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionExtended real number line, object, no evidence levelExtended real number …objectIntegral of a non-negative function, concept, no evidence levelIntegral of a non-neg…conceptMeasure space, concept, no evidence levelMeasure spaceconceptProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionStrongly measurable function, concept, no evidence levelStrongly measurable f…conceptSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionSupremum, definition, no evidence levelSupremumdefinitionEmpty set, object, no evidence levelEmpty setobjectFamily of sets, definition, no evidence levelFamily of setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionIntersection, definition, no evidence levelIntersectiondefinitionUnion, definition, no evidence levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionTopological space, definition, no evidence levelTopological spacedefinitionFunction, definition, no evidence levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionConvergence of a sequence in a metric space, definition, no evidence levelConvergence of a sequ…definitionOpen ball, definition, no evidence levelOpen balldefinitionDiameter of a set, definition, no evidence levelDiameter of a setdefinitionMetric topology, definition, no evidence levelMetric topologydefinitionEpsilon characterisation of Cauchy sequences, statement, no evidence levelEpsilon characterisat…statementComplete metric space, definition, no evidence levelComplete metric spacedefinitionOpen set, definition, no evidence levelOpen setdefinitionEquivalent metrics, definition, no evidence levelEquivalent metricsdefinitionEquivalent norms, definition, no evidence levelEquivalent normsdefinitionBanach space, definition, no evidence levelBanach spacedefinitionBochner integral, definition, no evidence levelBochner integraldefinition
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. Element definition
  2. Extended real number line object
  3. Integral of a non-negative function concept
  4. Measure space concept
  5. Proposition concept
  6. Real numbers object
  7. Set bounded above definition
  8. Strongly measurable function concept
  9. Set definition
  10. Conjunction definition
  11. Disjunction definition
  12. Implication definition
  13. Predicate definition
  14. Supremum definition
  15. Empty set object
  16. Family of sets definition
  17. Ordered pair definition
  18. Intersection definition
  19. Union definition
  20. Existential quantifier definition
  21. Universal quantifier definition
  22. Cartesian product definition
  23. Subset definition
  24. Union of a family of sets definition
  25. Uniqueness quantifier definition
  26. Binary relation definition
  27. Topological space definition
  28. Function definition
  29. Metric space definition
  30. Convergence of a sequence in a metric space definition
  31. Open ball definition
  32. Diameter of a set definition
  33. Metric topology definition
  34. Epsilon characterisation of Cauchy sequences statement
  35. Complete metric space definition
  36. Open set definition
  37. Equivalent metrics definition
  38. Equivalent norms definition
  39. Banach space definition
  40. Bochner integral definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Bochner_integral
About this page
TagENTNQSXA96
URIm2://ENTNQSXA96
Kinddefinition
WritingW0 stub
MSC28B05, 46G10
DefinesBochner integral
UsesMeasure space, Strongly measurable function, Integral of a non-negative function, Banach space
Changed2 November 2026

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