The first Radon-Nikodým
theorem for the Bochner integral was proven by Dunford and Pettis in 1940. In
1943, Phillips proved an extension of the Dunford and Pettis result. Then in
1968-69, three results appeared. One of these, due to Metivier, bears a direct
resemblence to the earlier Phillips theorem. The remaining two were proven
by Rieffel and seem to stand independent of the others. This paper is an
attempt to put these apparently diverse theorems in some perspective by
showing their connections, by simplifying some proofs and by providing
some modest extensions of these results. In particular, it will be shown that
the Dunford and Pettis theorem together with Rieffel’s theorem directly
imply Phillips’ result. Also, it will be shown that, with almost no sacrifice of
economy of effort, the theorems here can be stated in the setting of the Pettis
integral.
