ABSTRACT
This chapter collects some preliminary facts and concentrates on Hilbert spaces. It considers integration theory for measurable functions assuming their values in a Banach space. The chapter extends the notion of measurability of functions to Banach spaces-valued functions. It discusses some convergence theorems in analogy with the ones in the theory of the Lebesgue integral. The Bochner integral is an advanced topic for beginners. In a word the Bochner integral replaces R or C in the range by Banach spaces. The notion of weak measurablity corresponds to the separation of the real-valued functions into the positive part and the negative part. The notion of countably valuedness is used to guarantee that this operation can be completed within countably infinite steps.
