reference for representation of linear functionals of random variables

#1
I'm looking for a representation theorem for linear functionals of random variables. I'm not sure if Riesz–Markov–Kakutani does it, but it seems that something like that is going to do it. Can anyone point me to a theorem? A reference to a book discussing random variables at that level of technicality would be helpful, too. Thanks in advance.
 
#3
What I really needed to know about was Daniell Integrals. Halmos did not help. Found it in Ash, Robert B. (1972) Real Analysis and Probability, New York: Academic Press.