  The Bakry-Emery Ricci tensor of a metric-measure space (M,g,e^{-f}dv_{g})
plays an important role in both geometric measure theory and the study of
Hamilton's Ricci flow. Under a uniform positivity condition on this tensor and
with bounded Ricci curvature we show the underlying space has finite f-volume.
As a consequence such manifolds, including shrinking Ricci solitons, have
finite fundamental group. The analysis can be extended to classify shrinking
solitons under convexity or concavity assumptions on the measure function.
