This paragraph we follow the book of Rudin . On linear functionals, there is special relationship between integration and linear functionals. In , a vector space, for any positive measure , the mapping
example, let be the set of all continuous functions on the unit interval . Then
has two properties:
so it is a linear functional on . Moreover, this is a positive linear functional: if then .
Consider a segment and the class of all with and or support of is . So we get . It is important that the length of related to the values of the functional .
There is an important theorem of F. Riesz, this illustrates to above event
Theorem 1 (F. Riesz) Let be a closed subset of . Then every positive linear functional on there corresponds a finite positive Borel measure on such that
 W. Rudin, Real and Complex analysis, Mc.Graw-Hill, 1970.