There are nonequivalent geometries, nonequivalent groups finite and infinite, nonequivalent logics ( fregean and nofregean <http://www.formalontology.it/suszkor.htm>), even nonequivalent logicians;-)


**Are there nonequivalent randomnesses?** 


The main two theories we know dealing with randomness and probability is Kolmogorov Probability Theory ( by means of measure theory and Borel $\sigma$-algebras), and Bayesian a priori approach. Are they equivalent enough to say that they are the same in some deeper meaning?




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**@ Johannes Hahn** - no, I am not asking about non isomorphic probability spaces as it would be trivial. rather I as about possible probability theories as different as different are geometries euclidean and noneuclidean. 
The obvious generalization is to change linearity in the second axiom of probability ($P(A U B) = P(A) + P(B)$ when A,B are independent. 


In fact I mention about it after reading probability overview by Terence Tao <http://terrytao.wordpress.com/2010/01/01/254a-notes-0-a-review-of-probability-theory/> He wrotes:



> 
> probability theory is only “allowed”
>  to study concepts and perform
>  operations which are preserved with
>  respect to extension of the underlying
>  sample space.
> 
> 
> 


Which in my opinion is something very deep ( but I am only a hobbyist;-). So probably if You have initial probability space, and You need to extent it to describe some additional phenomena, You have have done some kind of morphisms between structures of first space and another wider one. Is there a **unique**, canonical or any defined way of doing this? May we perform this kind of extension always in the same way or there are different ways of doing that? Gives it any predictable and interesting structure?


**@sheldon-cooper** Bayesian approach to probability is sometimes seen as alternative ( not very well defined) to Kolmogorov axiomatic probability system, because it do not require given a priori probability space. For example in this approach we may say that probability that tomorrow will be could day ( say temp<-10) is defined, whilst in Kolmogorov approach You probably cannot define proper space ( because You cannot have equivalent population of Wednesdays which are tomorrow days, with different temperatures). Agree - when You have possibility to use properly defined probability space this two approach coincides. From Wikipedia: <http://en.wikipedia.org/wiki/Bayesian_probability>



> 
> Bayesian probability interprets the
>  concept of probability as "a measure
>  of a state of knowledge",[1] in
>  contrast to interpreting it as a
>  frequency or a physical property of a
>  system.
> 
> 
> 


**@Qiaochu Yuan** - of course randomness here is colloquialism. Yes You have right: maybe I just should ask about different probability theories, but note that non-euclidean geometries in analogy are just geometries but in different spaces, with some special properties. So in fact they share the same meaning of geometric set, figure, space, even so complicated objects as coordinate systems, and angle. But they have different relations between them. So I ask about something similar: different kinds of randomness which are in scope of probability theory but describes different relations between for example different classes of ways of extensions of probability spaces. If the last procedure changes anything in resultants;-) Agree - maybe this is not very interesting question. Maybe it would be more interesting in scope of algorithmic information theory and its randomness concept?



Are there nonequivalent randomnesses?
A different answer from the ones so far: Quantum randomness is another kind of randomness that is a generalization of traditional randomness, i.e. classical or non-quantum probability. I think that it fits the question because you could likewise say that non-Euclidean geometry, interpreted as not-necessarily-Euclidean geometry, is a generalization of Euclidean geometry.


A classical probability space is usually defined as a $\sigma$-algebra $\Omega$ with a normalized measure. From the Bayesian viewpoint the measure could equally well be called a "state". Now, a $\sigma$-algebra is the algebra of Boolean random variables with a certain set of axioms. But you can just as well write down axioms for $L^\infty(\Omega)$, the algebra of bounded complex random variables. In favorable cases, it is a commutative von Neumann algebra. In quantum probability you instead allow a non-commutative von Neumann algebra $\mathcal{M}$. Also, in standard quantum probability you keep the usual completed tensor product $\mathcal{M} \otimes \mathcal{N}$ as the model of a joint system. (Free probability theory is still quantum probability, but with a certain free product instead of a tensor product.) You also still have states, conditional states, joint states, correlations, generalized stochastic maps, etc.


Some of the variant models mentioned so far lead to different theorems, but generally give the same answers in combinatorial probability, questions like the birthday paradox or modeling games of chance. Quantum probability leads to a significantly different picture of combinatorial probability, generalizing the old one, but also allowing new answers such as violation of Bell's inequalities, covariance matrices that are Hermitian rather than real symmetric, new complexity classes such as BQP, etc.


Other variant models mentioned so far no longer give any answers for combinatorial probability, for instance models of forcing. But, part of the interest in probability is that it models real life. Amazingly, so does quantum probability; that was the central discovery of quantum mechanics when it was defined in the 1920s and 1930s.


