On a metric measure space, the authors consider a family of Lipschitz-Besov spaces defined only using the metric and the measure, and a family of Besov spaces defined using an auxiliary self-adjoint operator \(L\) and the associated heat semigroup. Under the assumptions about the heat kernel of \(L\) is stochastic complete, it is bounded in \(L^1\), it has Hölder regularity and a suitable upper bound holds, the authors prove a very interesting result: the identity of the two families of the function spaces.