Let be a nonempty set and be a -algebra on . Also, let be a non-negative measure defined on . Two complex valued functions and are said to be equal almost everywhere on (denoted as a.e. if The relation of being equal almost everywhere on defines an equivalence relation. It is a common practice in the integration theory to denote the equivalence class containing by itself. It is easy to see that if are equivalent and are equivalent, then are equivalent, and are equivalent. This naturally defines addition and multiplication among the equivalent classes of such functions. For a measureable , we define
called the essential supremum of on . Now we define,
Here the elements of are equivalence classes.
is the dual of if is -finite.
|Date of creation||2013-03-22 13:59:46|
|Last modified on||2013-03-22 13:59:46|
|Last modified by||ack (3732)|