1

I suddenly wondered if the change of the order of integral is valid even when its integral diverges. For the presence, I knew that Tonelli's theorem is exactly it (from Wikipedia: https://en.wikipedia.org/wiki/Fubini%27s_theorem#Tonelli.27s_theorem) but I can not quite understand.

Is the integrability not necessary for Tonelli theorem? That is, if "$(X,A,\mu)$ and $(Y,B,\nu)$ are $\sigma$-finite measure space and $f:X\times Y\to[0,\infty]$ is measurable" are only assumed, then can we change the order of integrals?

Another notes I found are written that the integral is finite and the change of order of integrals (Fubini theorem) is basically needed the integrality, doesn't it. I am confused because of these...

Could you tell me why Fubini like theorem holds even if the integrability is not assumed?

Thanks in advance.

user
  • 519
  • This is from wikipedia: "Fubini's theorem implies that the two repeated integrals of a function of two variables are equal if the function is integrable. Tonelli's theorem introduced by Leonida Tonelli (1909) is similar but applies to functions that are non-negative rather than integrable." – Giovanni Oct 13 '15 at 04:28
  • I couldn't understand also this sentence. That is, is Tonelli theorem imposed assumptions of Fubini theorem? – user Oct 13 '15 at 04:34
  • 1
    The statement you quoted in the yellow box is Tonelli's theorem and it is stated correctly. Indeed, you can apply Tonelli's theorem to nonnegative non integrable functions. – Giovanni Oct 13 '15 at 04:49

0 Answers0