Algebraic theories for extensive measurement are traditionally framed in terms of a binary relation ≲ and a concatenation (x,y) → xy. For situations in which the data is “noisy,” it is proposed here to consider each expression y ≲ x as symbolizing an event in a probability space. Denoting P(x,y) the probability of such an event, two theories are discussed corresponding to the two representing relations: with Axiomatic analyses are given, and representation theorems are proven in detail.