Contractive projections and conditional expectations

If ([Omega], [Sigma], [mu]) is a general measure space, where [Sigma] is a [delta]-ring, let Lp([Sigma]) be the corresponding Lebesgue space. One of the authors has defined and studied the properties of the (generalized) conditional expectation E[Lambda] on Lp([Sigma]) relative to a [delta]-ring [Lambda] [subset of] [Sigma] which reduces to the classical case when these [delta]-rings are [sigma]-algebras and [mu] is finite (cf. N. Dinculeanu, J. Multivariate Anal., 1 1971, 347-364). In this paper contractive projections on the space Lp([Sigma]), 1 <= p <= [infinity] are characterized in terms of the (generalized) conditional expectations under varying hypotheses for: (i) the general contractive projections, (ii) contractive positive projections, (iii) contractive averaging operators, and (iv) a brief analog of these operators on of X-valued functions, where X is a Banach space. In this treatment, when specialized classes of projections are concerned, the corresponding hypotheses on these operators are progressively weakened. This work extends most of the earlier studies on these problems, known to the authors, in the context of the Lebesgue spaces.