This is the third paper in a series in which we prove Thurston's conjectural duality between best Lipschitz maps and transverse measures. In the second paper we found a special class of best Lipschitz maps between hyperbolic surfaces (infinity harmonic maps), which induce dual Lie algebra valued transverse measures with support on Thurston's canonical lamination. The present paper examines these Lie algebra valued measures in greater detail. For any measured lamination we are led to define a Lie algebra valued measure and conversely every Lie algebra valued transverse measure arrises from this process. Furthermore, we show that such measures are infinitesimal earthquakes. This construction provides a natural correspondence between best Lipschitz maps and earthquakes.