Remark 2. The convergence of the sequence and domination by g {\displaystyle g} can be relaxed to hold only μ-almost everywhere provided the measure space (S, Σ, μ) is complete or f {\displaystyle f} is chosen as a measurable function which agrees μ-almost everywhere with the μ-almost everywhere existing pointwise limit. (These precautions are necessary, because otherwise there might exist a non-measurable subset of a μ-null set N ∈ Σ, hence f {\displaystyle f} might not be measurable.)