Math 2.0

Sequences, series, summability Theorem

Uniform limits commute with integration

Builds on Uniform convergenceRiemann integralMonotonicity of the integralRiemann's criterion for integrability

Statement

Let a<ba < b, and let fn ⁣:[a,b]→ℝf_n \colon [a, b] \to \mathbb{R} be Riemann integrable for each nn. If fn→ff_n \to f uniformly on [a,b][a, b], then ff is Riemann integrable on [a,b][a, b], and lim⁡n→∞∫abfn(x) dx=∫abf(x) dx.\lim_{n \to \infty} \int_a^b f_n(x) \, dx = \int_a^b f(x) \, dx.

Remarks

Let εn=sup⁡∣fn−f∣\varepsilon_n = \sup |f_n - f|. Then fn−εn≤f≤fn+εnf_n - \varepsilon_n \le f \le f_n + \varepsilon_n. So the upper and lower sums of ff differ by at most 2εn(b−a)2 \varepsilon_n (b - a) more than those of fnf_n. This gives integrability, by Riemann's criterion. By monotonicity, ∣∫abfn−∫abf∣≤εn(b−a)→0\big| \int_a^b f_n - \int_a^b f \big| \le \varepsilon_n (b - a) \to 0. Pointwise convergence is not enough. On [0,1][0, 1], the functions fn(x)=n2x(1−x2)nf_n(x) = n^2 x (1 - x^2)^n tend to 00 pointwise, but ∫01fn=n22n+2→∞\int_0^1 f_n = \frac{n^2}{2n + 2} \to \infty.

Reading path

To understand this page from the ground up, read these 18 pages in order. The foundations come first.

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Partition of an interval, definition, no evidence levelPartition of an inter…definitionSequence, definition, no evidence levelSequencedefinitionMesh of a partition, definition, no evidence levelMesh of a partitiondefinitionRiemann sum, definition, no evidence levelRiemann sumdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionRiemann integrable function, definition, no evidence levelRiemann integrable fu…definitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionRiemann integral, definition, no evidence levelRiemann integraldefinitionUpper and lower Darboux sums, definition, no evidence levelUpper and lower Darbo…definitionPointwise convergence, definition, no evidence levelPointwise convergencedefinitionDefinite integral, definition, no evidence levelDefinite integraldefinitionUpper and lower Darboux integrals, definition, no evidence levelUpper and lower Darbo…definitionUniform convergence, definition, no evidence levelUniform convergencedefinitionDarboux and Riemann integrals agree, statement, no evidence levelDarboux and Riemann i…statementMonotonicity of the integral, statement, no evidence levelMonotonicity of the i…statementRiemann's criterion for integrability, statement, no evidence levelRiemann's criterion f…statementUniform limits commute with integration, statement, no evidence levelUniform limits commut…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.
  • no evidence level
  1. Partition of an interval definition
  2. Sequence definition
  3. Mesh of a partition definition
  4. Riemann sum definition
  5. Real sequence definition
  6. Riemann integrable function definition
  7. Bounded sequence definition
  8. Limit of a sequence definition
  9. Riemann integral definition
  10. Upper and lower Darboux sums definition
  11. Pointwise convergence definition
  12. Definite integral definition
  13. Upper and lower Darboux integrals definition
  14. Uniform convergence definition
  15. Darboux and Riemann integrals agree statement
  16. Monotonicity of the integral statement
  17. Riemann's criterion for integrability statement
  18. Uniform limits commute with integration statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Uniform_convergence, To integrability
About this page
Tag0BWGW68B66
URIm2://0BWGW68B66
Kindstatement, theorem
WritingW0 stub
MSC40A30, 26A42
StatusThis copy of the site has no build data. Run m2 status 0BWGW68B66.
UsesUniform convergence, Riemann integral, Monotonicity of the integral, Riemann's criterion for integrability

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