In [AS], we de ned a notion of measure on the complexity class P (in the spirit of the work of Lutz [L92] that provides a notion of measure on complexity classes at least as large as E, and the work of Mayordomo [M] that provides a measure on PSPACE). In this paper, we show that several other ways of de ning measure in terms of covers and martingales yield precisely the same notion as in [AS]. ...

In this paper we prove almost sure convergence to the ball, in the Nikodym metric, of sequences of random Steiner symmetrizations of bounded Caccioppoli and bounded measurable sets, paralleling a result due to Mani-Levitska concerning convex bodies. AMS classification: 60D05, 52A40, 28A05

0. Introduction. Throughout this paper T will denote a measure preserving transformation on a cr-finite infinite measure space (X, (B, m) which is point isomorphic with the Lebesgue measure space of the real line. Unless otherwise stated, T will be one-one. Equations involving functions or sets will always be interpreted modulo sets of measure zero. T is said to be ergodic if T~1E = E, ££(B, im...