Algorithmically random functions and effective capacities

We continue the investigation of algorithmically random functions and closed sets, and in particular the connection with the notion of capacity. We study notions of random continuous functions given in terms of a family of computable measures called symmetric Bernoulli measures. We isolate one particular class of random functions that we refer to as online random functions $F$, where the value of $y(n)$ for $y = F(x)$ may be computed from the values of $x(0),\dots,x(n)$. We show that random online functions are neither onto nor one-to-one. We give a necessary condition on the members of the ranges of online random functions in terms of initial segment complexity and the associated computable capacity. Lastly, we introduce the notion of online \emph{partial} Martin-Löf random function on $2^ω$ and give a family of online partial random functions the ranges of which are precisely the random closed sets introduced by Barmpalias, Brodhead, Cenzer, Dashti, and Weber.

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