The Bochner Integral (lehrbücher Und Monographien Aus Dem Gebiete Der Exakten Wissenschaften)

The theory of the Lebesgue integral is still considered as a difficult theory, no matter whether it is based the concept of measure or introduced by other methods. The primary aim of this book is to give an approach which would be as intelligible and lucid as possible. Our definition, produced in Chapter I, requires for its background only a little of the theory of absolutely convergent series so that it is understandable for students of the first undergraduate course. Nevertheless, it yields the Lebesgue integral in its full generality and, moreover, extends automatically to the Bochner integral (by replacing real coefficients of series by elements of a Banach space). It seems that our approach is simple enough as to eliminate the less useful Riemann integration theory from regular mathematics courses. Intuitively, the difference between various approaches to integration may be brought out by the following story on shoemakers. A piece of leather, like in Figure 1, is given. The task consists in measuring its area. There are three shoemakers and each of them solves the task in his own way. A B Fig. 1 The shoemaker R. divides the leather into a finite number of vertical strips and considers the strips approximately as rectangles. The sum of areas of all rectangles is taken for an approximate area of the leather (Figure 2). If he is not satisfied with the obtained exactitude, he repeats the whole procedure, by dividing the leather into thinner strips. iii_iii......Page 1 iv_iv......Page 2 v_vii......Page 3 1_7......Page 6 8_14......Page 13 15_22......Page 20 23_36......Page 28 37_39......Page 42 40_52......Page 45 53_64......Page 58 65_82......Page 70 83_90......Page 88 91_105......Page 96 106_113......Page 111 114_163......Page 119 164_182......Page 169 183_200......Page 188 201_210......Page 206 211_221......Page 216 222_226......Page 227 227_231......Page 232 232_233......Page 237 By Jan Mikusiński. Originally Published In Basel And Stuttgart By Birkhäuser Bibliography: P. [232]-233