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We use ideas from topological dynamics (amenability), combinatorics(structural Ramsey theory) and model theory (Fra\" {i}ss\' e limits) to studyclosed amenable subgroups $G$ of the symmetric group $S_\infty$ of a countableset, where $S_\infty$ has the topology of pointwise convergence. We construct$G$-invariant measures on the universal minimal flows associated with thesegroups $G$ in, moreover, an algorithmic manner. This leads to an identificationof the generic elements, in the sense of being Martin-L\" of random, of theseflows with respect to the constructed invariant measures. Along these lines westudy the random elements of $S_\infty$, which are permutations that transformrecursively presented universal structures into such structures which areMartin-L\" of random.

Martin-Lof randomness, invariant measures and countable homogeneous structurs