  We define and prove the existence of a fractional Brownian motion indexed by
a collection of closed subsets of a measure space. This process is a
generalization of the set-indexed Brownian motion, when the condition of
independance is relaxed. Relations with the Levy fractional Brownian motion and
with the fractional Brownian sheet are studied. We prove stationarity of the
increments and a property of self-similarity with respect to the action of
solid motions. Moreover, we show that there no "really nice" set indexed
fractional Brownian motion other than set-indexed Brownian motion. Finally,
behavior of the set-indexed fractional Brownian motion along increasing paths
is analysed.
