i have these sentences in my lecture notes :
"A problem that is important for positive linear functionals, is that the supremum over many elements may not be the same as the linear functional evaluated at the maximal elements using the partial orderning.
This leads to problems when trying to generalize measure and integrals to the noncommunitative case."
now my questions are : what does it mean of commutativity or noncommutativity of measures or integrals ?
how this noncommutativity can be related to the problem of supremum ?