In Henstock-Kurzweil integral in the proof of Fubini theorem you need a strictly positive integrable function for rectangles of infinite volume. How to deal with such situation in general setting . Lebesgue integral requires no condition for proof. Especially how to get such a function for Henstock-Kurzweil integral in general setting or in the setting of locally compact topological spaces, and complete separable metrizable spaces?
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A partial answer is that for Fubini's theorem the integrabilty of a mapping over a product forces existence of a positive map whose riemann sums are bounded. So for fubini's theorem the existence of a positive integrable function is unnecessary. but one needs to find a similar breakthrough for Monotone convergence theorem. The existence of such a function is found necessary only when the domain has infinite measure for example R. |
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