Math 2.0

Real functions Theorem

Fubini's theorem

Builds on Double integralIterated integralContinuity in R^n

Statement

Let R=[a,b]×[c,d]R = [a, b] \times [c, d], and let f ⁣:R→ℝf \colon R \to \mathbb{R} be Riemann integrable on RR. Suppose that y↦f(x,y)y \mapsto f(x, y) is integrable on [c,d][c, d] for each x∈[a,b]x \in [a, b], and that x↦f(x,y)x \mapsto f(x, y) is integrable on [a,b][a, b] for each y∈[c,d]y \in [c, d]. Then ∬Rf dA=∫ab∫cdf(x,y) dy dx=∫cd∫abf(x,y) dx dy.\iint_R f \, dA = \int_a^b \int_c^d f(x, y) \, dy \, dx = \int_c^d \int_a^b f(x, y) \, dx \, dy. In particular, this holds for each continuous function ff on RR.

Remarks

The left side is the double integral, and the other two are iterated integrals. The Lebesgue version: let ff be Lebesgue measurable on ℝn×ℝm\mathbb{R}^n \times \mathbb{R}^m with ∫∣f∣<∞\int |f| < \infty. Then the double integral equals both iterated integrals, where the inner integrals exist for almost all values of the outer variable. Without integrability of ∣f∣|f|, the iterated integrals can differ. For non-negative functions, Tonelli's theorem removes the integrability hypothesis. Euler knew the case of continuous functions. Lebesgue proved a version for bounded measurable functions in 1904, and Fubini proved the version for integrable functions in 1907.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Fubini%27s_theorem, Riemann integrals
  2. Web https://en.wikipedia.org/wiki/Fubini%27s_theorem, For integrable functions
  3. Web https://en.wikipedia.org/wiki/Fubini%27s_theorem, History
About this page
Tag0R0T1VWH5X
URIm2://0R0T1VWH5X
Aliasesfubini-theorem
Kindstatement, theorem
WritingW1 stated
MSC26B15
StatusThis copy of the site has no build data. Run m2 status 0R0T1VWH5X.
UsesDouble integral, Iterated integral, Continuity in R^n
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