We continue the investigation of analytic spaces from the perspective of computable structure theory. We show that if p ≥ 1 is a computable real, and if Ω is a nonzero, non-atomic, and separable measure space, then every computable presentation of Lᵖ(Ω) is computably linearly isometric to the standard computable presentation of Lᵖ[0, 1]; in particular, Lᵖ[0, 1] is computably categorical. We also show that there is a measure space Ω that does not have a computable presentation even though Lᵖ(Ω) does for every computable real p ≥ 1.