Algorithmic randomness of continuous functions

Abstract

We investigate notions of randomness in the space $${{\mathcal C}(2^{\mathbb N})}$$ of continuous functions on $${2^{\mathbb N}}$$ . A probability measure is given and a version of the Martin-Lof test for randomness is defined. Random $${\Delta^0_2}$$ continuous functions exist, but no computable function can be random and no random function can map a computable real to a computable real. The image of a random continuous function is always a perfect set and hence uncountable. For any $${y \in 2^{\mathbb N}}$$ , there exists a random continuous function F with y in the image of F. Thus the image of a random continuous function need not be a random closed set. The set of zeroes of a random continuous function is always a random closed set.