  This paper deals with measurable stationary symmetric stable random fields
indexed by R^d and their relationship with the ergodic theory of nonsingular
R^d-actions. Based on the phenomenal work of Rosinski(2000), we establish
extensions of some structure results of stationary S alpha S processes to
S\alpha S fields. Depending on the ergodic theoretical nature of the underlying
action, we observe different behaviors of the extremes of the field.
