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Contractions with Fixed Points and Conditional Expectation

Let (Ω, α, μ) be a σ-finite measure space. By Lp(Ω, α, μ) or Lp for short we denote the usual Banach space of pth power μ-integrable functions on Ω if 1≤p<+ ∞ and μ-essentially bounded functions on Ω, if p= +∞. In section (2) we characterize conditional expectation, by a method different than those used previously. Modulus of a given contraction is discussed in section (3). If the given contraction has a fixed point, then its modulus has a simple form (theorem 3.2).