Z.Wahrscheinlichkeitstheorie verw. Geb. 21, 154-166 (1972) 9 by Springer-Verlag 1972 The Exit Measure of a Supermartingale HANS FOLLMER Let X = (Xt)t>=o be a nonnegative right continuous supermartingale relative to an increasing family (~t)t>=o of a-fields on a probability space (f2, ~, P). It is well known that X has a boundary function X~o." = lira Xt, and if X is a uniformly integrable martingale then the boundary function actually determines X in the sense that X~ is the conditional expectation of X~ with respect to ~. In general however the boundary function may provide very little information on X. In this paper we give a characterization of X by its terminal behaviour which applies to the general case. Here the terminal behaviour is specified by a certain measure, the exit measure of X. It reduces to the measure X~ dP whenever X is a uniformly integrable martingale. As an illustration consider classical potential theory on the unit disc where the supermartingale X arises by observing a superharmonic function u > 0 along Brownian motion paths. Any such u is characterized by a finite measure on the closed unit disc (Poisson-Riesz representation), which may be viewed as the terminal distribution of a certain Markov process associated to u, the so called u-path process introduced by Doob in [31. Analytically, this exit measure is just the Choquet measure which represents u as a mixture of extremal rays in the cone of superharmonic functions > 0. This example will serve as a guideline for our discussion. In Section 1 we construct a probability measure pX on the a-field N of previsible sets in O.. = f2 x [0, oo] such that 1 px[Ax(t, oo]]= E[Xo. ] E[Xt;A 1 (A~, t>O). The second coordinate of ~ serves as a lifetime. The measure pX generalizes the notion of a u-path process; a more explicit discussion of this can be found in E8]. The construction is based on the Ito-Watanabe factorization of X into a local martingale and a decreasing process. This factorization allows to define pX consistently on an increasing sequence of a-fields in P, similarly to the construc- tion of e-subprocesses of a Hunt process in [5]. In order to extend pX to ~ we introduce the following regularity assumption on the underlying a-fields: (~) is the right continuous modification of a standard system (fit~ The notion of a standard system is essentially due to Parthasarathy in [91. It means that (i) each a-field ~t ~ is standard Borel and that (ii) decreasing sequences of atoms have a non void intersection (cf. Appendix). Condition (ii) could be dropped if we were ready to replace ~ by an inverse limit space. Path spaces of type D(0, oo) satisfy (i) but not (ii). However, and this remark is due to Meyer, if we allow for 'explosion