The Measurability And Integral Of A Function With Values In Locally Convex Separated Spaces

In this thesis, the concepts of(?) element-measurability and T-measurability are de-fined for a function defined on a complete finite measure space with values in locally convex separated spaces. Some equivalent conditions for these two measurability of a vector func-tion are obtained and it is pointed out that the concepts of measurability are equivalent with the concept ofμ- measurability in normed linear spaces. Then, Pettis' measurability theorem is generalized to locally convex separated spaces. From this, some basic properties of(?) element-measurable(T-measurable) vector functions sequences are discussed. Finally, the concept of (?)- integrability(T-integrability) is defined for a (?) element-measurable (T-measurable) vector function, some properties of (?)- integral(T-integral) are given which are analogous to those for the Bochner integral and the equivalent condition about T-integrability are explored.