I don't understand why this example doesn't satisfy the conditions of the Tonelli result. It is said that the space are not $\sigma$-finite. But isn't the counting measure and the Lebesgue measure $\sigma$-finite?
2026-05-11 03:41:01.1778470861
Counter-example to tonelli's theorem.
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Counting measure is not sigma finite. We cannot write $[0,1]$ as a countable union of finite sets.