An Optimal Superfarthingale and Its Convergence over a Computable Topological Space

We generalize the convergenece of an optimal semimeasure to a real probability in algorithmic probability by using game-theoretic probability theory and the theory of computable topology. Two lemmas in the proof give as corollary the existence of an optimal test and an optimal integral test, which are important from the point of view of algorithmic randomness. We only consider an SCT3 space, where we can approximate the measure of an open set. Our proof of almost-sure convergence to the real probability by a superfarthingale indicates why the convergence in Martin-Löf sense does not hold.