Editing Fubini's theorem (section)

==Proofs==

Proofs of the Fubini and Tonelli theorems are necessarily somewhat technical, as they have to use a '''hypothesis''' related to '''σ-finiteness'''. Most proofs involve building up to the full theorems by proving them for increasingly complicated functions, with the steps as follows.
# Use the fact that the measure on the product is multiplicative for rectangles to prove the theorems for the characteristic functions of rectangles.
# Use the condition that the spaces are σ-finite (or some related condition) to prove the theorem for the characteristic functions of measurable sets. This also covers the case of simple measurable functions (measurable functions taking only a finite number of values).
# Use the condition that the functions are measurable to prove the theorems for positive measurable functions by approximating them by simple measurable functions. This proves Tonelli's theorem.
# Use the condition that the functions are integrable to write them as the difference of two positive integrable functions and apply Tonelli's theorem to each of these. This proves Fubini's theorem.

===Riemann integrals===
For [[Riemann integral]]s, Fubini's theorem is proven by refining the partitions along the x-axis and y-axis as to create a joint partition of the form <math>[x_i,x_{i+1}] \times [y_j,y_{j+1}]</math>, which is a partition over <math>X\times Y</math>. This is used to show that the double integrals of either order are equal to the integral over <math>X\times Y</math>.