A probabilistic secret sharing scheme is a joint probability distribution of the shares and the secret together with a collection of secret recovery functions. The study of schemes using arbitrary probability spaces and unbounded number of participants allows us to investigate their abstract properties, to connect the topic to other branches of mathematics, and to discover new design paradigms. A scheme is perfect if unqualified subsets have no information on the secret, that is, their total share is independent of the secret. By relaxing this security requirement, three other scheme types are defined. Our first result is that every (infinite) access structure can be realized by a perfect scheme where the recovery functions are non-measurable. The construction is based on a paradoxical pair of independent random variables which determine each other. Restricting the recovery functions to be measurable ones, we give a complete characterization of access structures realizable by each type of the schemes. In addition, either a vector-space or a Hilbert-space based scheme is constructed realizing the access structure. While the former one uses the traditional uniform distributions, the latter one uses Gaussian distributions, leading to a new design paradigm.