In the book that I'm reading (see the title), the author define an integral and its properties on a function having for domain a complete measure space and for codomain the Real line equipped with the Borel sigma-algebra. I wanted to know what is happening to the value of the integral and its properties if I change the sigma algebra of the domain and/or the sigma algebra of the codomain for another random sigma algebra. How does the choice of a sigma-algebra on the domain and/or the codomain influence the integral and its properties? It been 2 days i am searching on the internet and I see nothing on the subject. For example, does the integral of a function having the lebesgue measurable sigma-algebra for the domain equal the integral of the same function if the domain is equipped with the Borel sigma-algebra (since one is the sigma-algebra completion of the other)? (excuse my english, it is not my first language). Also, with what sigma algebra does the lebesgue integral equal the Riemann integral (in my textbook it says borel sigma-algebra at the codomain and borel sigma algebra at the domain but i am not sure if it is the only way possible)
2026-03-29 06:33:44.1774766024
Gail S. Nelson' *A User-Friendly Introduction to Lebesgue Measure and Integration* Confusion about general measure integration
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