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For which computable metric spaces does there exist a computable presentation of the corresponding L1 space of real-valued functions, and is there an algebraic or topological invariant that completely characterizes this property?

Related: Urysohn's theorem on computable Polish spaces, Pour-El and Richards theorem on computability in Banach spaces, Hoyrup and Rojas theorem on effective Borel measures

The paper on computable presentations of separable Lebesgue spaces raises a foundational question that remains unresolved: given an arbitrary computable metric space X equipped with a computable measure, can we always construct a computable presentation of the associated L1 function space over X? A computable presentation means an explicit, algorithmic enumeration of a dense sequence along with computable distance information, allowing computability theory to interact meaningfully with classical analysis. While specific cases such as the unit interval with Lebesgue measure are understood, the general landscape of which measure spaces yield computable L1 presentations and which do not remains murky, and no clean characterization is known.

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