Abstract
AbstractWe identify simple universal properties that uniquely characterize the Lebesgue spaces. There are two main theorems. The first states that the Banach space , equipped with a small amount of extra structure, is initial as such. The second states that the functor on finite measure spaces, again with some extra structure, is also initial as such. In both cases, the universal characterization of the integrable functions produces a unique characterization of integration. We use the universal properties to derive some of the basic elements of integration theory. We also state universal properties characterizing the sequence spaces and , as well as the functor taking values in Hilbert spaces.
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