Given a complete metric measure space whose measure is doubling and supports an ∞- Poincar´e inequality, and a bounded domain Ω in such a space together with a Lipschitz function f : ∂Ω → R, we show the existence and uniqueness of an ∞-harmonic extension of f to Ω. To do so, we show that there is a metric that is bi-Lipschitz equivalent to the original metric, such that with respect to this new metric the metric space satisfies an ∞- weak Fubini property and that a function which is ∞-harmonic in the original metric must also be ∞-harmonic with respect to the new metric. We also show that if the metric on the metric space satisfies an ∞-weak Fubini property, then the notion of ∞-harmonic functions coincide with the notion of AMLEs proposed by Aronsson. The notion of ∞-harmonicity is in general distinct from the notion of strongly absolutely minimizing Lipschitz extensions found in [13, 25, 26], but coincides when the metric space supports a p-Poincar´e inequality for some finite p ≥ 1.
