Find a concept
Start typing to search the mathematical index.
Theorem
The integral of exp minus a positive multiple of the square is the square root of pi divided by that multiple.
For , the Gaussian integral is
Its product form is , by Tonelli's theorem.
The integral for is finite and positive. Its square equals . In polar coordinates this becomes . Taking the positive square root gives ; substitution gives the general case.
Tonelli's theorem: Let and be -finite measure spaces, and let be -measurable. Then the functions
are measurable and
where all three integrals are allowed to be .
Here is the product measure on the Cartesian product, and the integrals are Lebesgue integrals of a nonnegative measurable function. Tonelli’s theorem is the nonnegative-function counterpart of Fubini's theorem, which applies under an absolute integrability hypothesis.