Measure theory is the study of measures and their siamese twins, integrals. Its central position in analysis is partly explained because it provides us with large families of Banach spaces. In this seminar I will try to explain the why, the what and the how of categorifying measure theory. It will consist mostly in laying the groundwork so as to be able to explain the construction of the category of \"categorified integrable functionsänd the integral functor. Time permitting, I will also give a categorified Radon- Nikodym theorem. In a second, future seminar, I will give categorified versions of other basic theorems of measure theory (e.g. Fubini and Fubini-Tonelli on the equality of iterated integrals) and explain some new phenomena peculiar to the categorified setting and with no counterpart in ordinary measure theory.