MathLiveFubini's theorem
For an integrable function on a product of two measure spaces, the double integral over the product equals either iterated integral, so the order of integration can be swapped. It is used to evaluate double integrals by reducing them to one-variable integrals.
\int_{X\times Y} f(x,y)\,\mathrm{d}(x,y)=\int_X\left(\int_Y f(x,y)\,\mathrm{d}y\right)\,\mathrm{d}x=\int_Y\left(\int_X f(x,y)\,\mathrm{d}x\right)\,\mathrm{d}yWhat the symbols mean
| the integrable function of the two variables x and y | |
| the first measure space (domain of x) | |
| the second measure space (domain of y) | |
| the product measure element over X \times Y | |
| the measure element for integration over X | |
| the measure element for integration over Y |
About this formula
- FieldMathematics
- TopicMultivariable calculus and measure theory
- LevelUniversity
- Also calledTonelli-Fubini theorem, Fubini-Tonelli theorem, double integral iterated integral, swap order of integration, changing order of integration, iterated integrals equal double integral
- SourceWikidata Q1149022
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