The Ces\`aro function spaces Cesp​=[C,Lp], 1≤p≤∞, have
received renewed attention in recent years. Many properties of [C,Lp] are
known. Less is known about [C,X] when the Ces\`aro operator takes its values
in a rearrangement invariant (r.i.) space X other than Lp. In this paper
we study the spaces [C,X] via the methods of vector measures and vector
integration. These techniques allow us to identify the absolutely continuous
part of [C,X] and the Fatou completion of [C,X]; to show that [C,X] is
never reflexive and never r.i.; to identify when [C,X] is weakly sequentially
complete, when it is isomorphic to an AL-space, and when it has the
Dunford-Pettis property. The same techniques are used to analyze the operator
C:[C,X]→X; it is never compact but, it can be completely continuous.Comment: 21 page