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We extend a recent result of Tim Austin (see arXiv:0905.0515) to the L^1setting, thus providing a general version of the Birkhoff ergodic theorem forfunctions taking values in nonpositively curved spaces. In this setting, thenotion of a Birkhoff sum is replaced by that of a barycenter along the orbit.The construction of an appropriate barycenter map is the core of this note. Inparticular, we solve a problem raised by K.-T. Sturm showing that localcompactness for the underlying space is superfluous for the construction (thisextends a result of A. Es-Sahib and H. Heinich). As a byproduct of ourconstruction, we prove a fixed point theorem for actions by isometries on aBuseman space.

An L^1 ergodic theorem with values in a nonpositively curved space via a canonical barycenter map