Is the change of the integral order valid even when its integral diverges? - Integrability assumption in Tonelli theorem-

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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.