A convergence property for conditional expectation (Q1121592)

Let (X,\({\mathcal B},\mu)\) be a probability space, \({\mathcal A}^ a \)subalgebra of \({\mathcal B}\), \[ L({\mathcal A},{\mathcal B})=\{f\in L^ 1({\mathcal B})| \exists H_ f\in L^ 1({\mathcal B}:\quad | f| \leq H_ f\quad a.e.\},\quad T(t)=| f-E(f| {\mathcal B})|, \] and let M(f) be the g.l.b. of ``functions'' in \(L^ 1({\mathcal B})\) which dominate f a.e.. Then, for \(f\in L({\mathcal A},{\mathcal B})\), \(M(T^ n(f))\to 0\) a.e. and \(\sum^{\infty}_{1}T^ n(f)\) converges in \(L^ 1({\mathcal B})\), more precisely, \[ \sum^{\infty}_{1}E(T^ n(f))\leq 4M(| f|). \] If \({\mathcal A}\) is trivial, then \(T^ n(f)\to 0\) in \(L^{\infty}({\mathcal B})\).