In his notes on free probability, Terence Tao describes a general approach to non-commutative probability which prioritizes the algebra of random variables above the sample space; I find this conceptually appealing. I would be interested in finding a reference which develops this theory, even if only in the commutative case, to the point where one can reproduce standard probabilistic and measure-theoretic results (e.g. the SLLN, the central limit theorem), and I would also be interested in applications to a measure-space-free statement and proof of an ergodic theorem.
Motivation: A problem on a recent problem set of mine has convinced me that the measure-theoretic and probabilistic apparatus I'm familiar with would be more flexible if I didn't have to think about sample spaces. I am also interested in having a probabilistic language that adapts to quantum probability more readily.