Lp-Continuity of noncommutative conditional expectations

Let (M,τ) be a noncommutative probability space, (Mn)n≥1 a sequence of von Neumann subalgebras of M and N a von Neumann subalgebra of M. We introduce the notions of μ-approach and orthogonal approach for (Mn)n≥1 and prove that for any x∈Lp(M)(1≤p<∞) if and only if (Mn)n≥1 τ-approaches and orthogonally approaches N.