Vol. 38, No. 2, 1971

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Radon-Nikodým theorems for the Bochner and Pettis integrals

S. Moedomo and J. Jerry Uhl, Jr.

Vol. 38 (1971), No. 2, 531–536

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.

Mathematical Subject Classification 2000
Primary: 28A45
Secondary: 46E40, 46G10
Received: 28 January 1971
Published: 1 August 1971
S. Moedomo
J. Jerry Uhl, Jr.