Fubinisatsen

Fubinisatsen, also known as Fubini's theorem, is a fundamental result in measure theory and calculus that deals with the interchange of the order of integration for multiple integrals. It states that under certain conditions, the iterated integral of a function over a product space is equal to the integral of the function over the entire product space.

In simpler terms, if you have a function of several variables, say f(x, y), and you want to calculate its double integral over a rectangular region, Fubinisatsen tells you that you can integrate with respect to x first and then y, or with respect to y first and then x, and you will get the same result. The conditions for the theorem to hold are crucial. Typically, the function must be non-negative or integrable (meaning its absolute value has a finite integral).

The theorem has significant implications in various fields. In probability theory, it allows for the calculation of marginal distributions from joint distributions. In physics and engineering, it simplifies the computation of volumes, masses, and other quantities involving multi-dimensional integrals. Without Fubinisatsen, many calculations in these areas would be significantly more complex or even intractable. The name "Fubini" comes from the Italian mathematician Guido Fubini, who first stated a version of this theorem.