Daniell Integral And Expectation Language In Probability

In this paper we introduce the Daniell Integral and discuss the applications of this integral in probability. The applications provide us with a method that begins with expectations rather than probabilities. This method will be called "method via expectations", and when following this method to discuss and study probability problems,we say we are using "Expectation Language".Daniell Integral is a special mode of integral known from the Lebesgue's mode. It treats integral as a functional and so it is relatively more abstract. But it provides a new visual angle to look on integrals, and on expectations in probability theory. Its applications in probability theory was first introduced in 1920s by Daniell (1921) and Wiener (1923), (see[7][24][25]), but then almost being forgotten after Kolmogorov's Axioms. In recent years Peng introduced a new method to define nonlinear expectations via Daniell Integrals (see [18][19][20]).Since "Probability Language" is so familiar to us, it can not but influences our thoughts. Thus the author thinks it necessary to introduce the Daniell Integral systemically and discuss the characteristics of the "Expectation Language".There are five chapters in this paper.The first chapter is an introduction. In chapter 2, we introduce the Lebesgue's integral and W.H.Young's integral. The former is a theoretical foundation of the "probability language" compared with the Daniell Integral; and the later, as will be seen in the following chapter, offered the main methods for Daniell Integral.