Math 2.0

Measure and integration Product measures Theorem

Tonelli's theorem

Builds on Fubini's theoremIterated integral

Statement

Let f ⁣:ℝn×ℝm→[0,∞]f \colon \mathbb{R}^n \times \mathbb{R}^m \to [0, \infty] be Lebesgue measurable. Then x↦∫ℝmf(x,y) dyx \mapsto \int_{\mathbb{R}^m} f(x, y) \, dy and y↦∫ℝnf(x,y) dxy \mapsto \int_{\mathbb{R}^n} f(x, y) \, dx are measurable functions with values in [0,∞][0, \infty], and ∫ℝn×ℝmf d(x,y)=∫ℝn∫ℝmf(x,y) dy dx=∫ℝm∫ℝnf(x,y) dx dy.\int_{\mathbb{R}^n \times \mathbb{R}^m} f \, d(x, y) = \int_{\mathbb{R}^n} \int_{\mathbb{R}^m} f(x, y) \, dy \, dx = \int_{\mathbb{R}^m} \int_{\mathbb{R}^n} f(x, y) \, dx \, dy. The three values may be +∞+\infty.

Remarks

Tonelli's theorem has the same conclusion as Fubini's theorem. It replaces integrability of ∣f∣|f| by the hypothesis f≥0f \ge 0. In practice, one applies Tonelli's theorem to ∣f∣|f| to check that ∫∣f∣<∞\int |f| < \infty, and then applies Fubini's theorem to ff. More generally, the theorem holds for non-negative measurable functions on a product of two σ\sigma-finite measure spaces. For counting measure on ℕ\mathbb{N}, it says that ∑x∑yaxy=∑y∑xaxy\sum_x \sum_y a_{xy} = \sum_y \sum_x a_{xy} when all axy≥0a_{xy} \ge 0. Leonida Tonelli published this variation of Fubini's theorem in 1909.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptIterated integral, definition, no evidence levelIterated integraldefinitionLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptContinuity in R^n, definition, no evidence levelContinuity in R^ndefinitionDouble integral, definition, no evidence levelDouble integraldefinitionFubini's theorem, statement, no evidence levelFubini's theoremstatementTonelli's theorem, statement, no evidence levelTonelli's theoremstatement
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  1. Euclidean space object
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  4. Iterated integral definition
  5. Limit of a function of several variables concept
  6. Continuity in R^n definition
  7. Double integral definition
  8. Fubini's theorem statement
  9. Tonelli's theorem statement

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Fubini%27s_theorem, Tonelli's theorem for non-negative measurable functions
  2. Web https://en.wikipedia.org/wiki/Fubini%27s_theorem, History
About this page
Tag067SYD2J4B
URIm2://067SYD2J4B
Aliasestonelli-theorem
Kindstatement, theorem
WritingW0 stub
MSC28A35
StatusThis copy of the site has no build data. Run m2 status 067SYD2J4B.
UsesFubini's theorem, Iterated integral

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