Basically this says that you can switch the of integrals, or integrate over the product space as long as everything is positive and the spaces are -finite. Do note that we allow the functions to take on the value of infinity with the standard conventions used in Lebesgue integration. That is, , so that if a function is infinite on a set of measure 0, then this does not contribute anything to the value of the integral. See the entry on extended real numbers for further discussion.
Theorem (Tonelli for sums).
Suppose that for all , then
In the above theorem we have used as our set for simplicity and familiarity of notation. If you would have an uncountable number of non-zero elements then all the sums would be infinite and the result would be trivial. So the theorem for arbitrary sets just reduces to the above case.
- 1 Gerald B. Folland. . John Wiley & Sons, Inc., New York, New York, 1999
|Date of creation||2013-03-22 14:15:49|
|Last modified on||2013-03-22 14:15:49|
|Last modified by||jirka (4157)|