6 CHapTER L PRELIMINARY in vals (cubes). underlining Euclidean geometry,on the other hand,put strong restrictions onto the local behavior of emanian integral represents theare e curve,th strips.Le horizontal strip may spread everywhere,however,it turns out to be a sweet surprise.As the local behavior of th in consideratio e care This viewpoint dramatically enlarges the range of integrable functions.The corresponding inte gral theory now boils down to the definition of the measure,and the rest follows alm nost naturally Another great advantage of reat convenience in subiect such as probability theory. -wise,in this course we shall provide the following generalization: length,.area,volume,…→measure continuous functions =measurable functions Riemannian integral=Lebesgue integral n the following,we sketch some important historical moments of the development for the real6 CHAPTER 1. PRELIMINARY The basic idea of Riemannian integral is to divide the domain of definition into small intervals (cubes for higher dimensions). These neighboring intervals (cubes), on the one hand, rely on the underlining Euclidean geometry, on the other hand, put strong restrictions onto the local behavior of integrable functions. (cannot oscillate too much, thus leading to the continuity to some extent) The geometric meaning of the Riemannian integral represents the area under the curve, thus Riemann’s way of integration, roughly speaking, is to approximate the area by dividing the region into vertical strips. Lebesgue’s viewpoint is to view the region by horizontal strips. At a first glance, each horizontal strip may spread everywhere, however, it turns out to be a sweet surprise. As the local behavior of the function in consideration is not so critical, and what really matters now is the set of the form {f ≥ c}, which motivates the careful definition of its measure (strictly speaking, in this book by measure we mean Lebesgue measure). This viewpoint dramatically enlarges the range of integrable functions. The corresponding inte￾gral theory now boils down to the definition of the measure, and the rest follows almost naturally. Another great advantage of Lebesgue’s integral theory is that it is not restricted only to the inte￾gration on Euclidean space. It can equally be transplanted to any abstract measure space, yielding great convenience in subject such as probability theory. We shall see the above counter-example holds true in the sense of Lebesgue’ integration. Namely, the Dirichlet function is Lebesgue integrable and our hope that limn→∞ R [0,1] fn(x)dx = R [0,1] D(x)dx becomes true. Vocabulary-wise, in this course we shall provide the following generalization: length, area, volume, ... =⇒ measure continuous functions =⇒ measurable functions Riemannian integral =⇒ Lebesgue integral In the following, we sketch some important historical moments of the development for the real analysis