The study of probabilistic secret sharing schemes using arbitraryprobability spaces and possibly infinite number of participants lets us

investigate abstract properties of such schemes. It highlights important

properties, explains why certain definitions work better than others,

connects this topic to other branches of mathematics, and might yield new

design paradigms.

A {\\em probabilistic secret sharing scheme} is a joint probability

distribution of the shares and the secret together with a collection of {\\em

secret recovery functions} for qualified subsets. The scheme is measurable

if the recovery functions are measurable. Depending on how much information

an unqualified subset might have, we define four scheme types: {\\em

perfect}, {\\em almost perfect}, {\\em ramp}, and {\\em almost ramp}. Our main

results characterize the access structures which can be realized by schemes

of these types.

We show that every access structure can be realized by a non-measurable

perfect probabilistic scheme. The construction is based on a paradoxical

pair of independent random variables which determine each other.

For measurable schemes we have the following complete characterization. An

access structure can be realized by a (measurable) perfect, or almost

perfect scheme if and only if the access structure, as a subset of the

Sierpi\\\'nski space $\\{0,1\\}^P$, is open, if and only if it can be realized

by a span program. The access structure can be realized by a (measurable)

ramp or almost ramp scheme if and only if the access structure is a

$G_\\delta$ set (intersection of countably many open sets) in the

Sierpi\\\'nski topology, if and only if it can be realized by a Hilbert-space

program.