In the paper the basic concepts of extended probability theory areintroduced. The basic idea: the concept of an event as a subset of \Omega isreplaced with the concept of an event as a partition. The partition is any setof disjoint non-empty subsets of \Omega (i.e. partition=subset+itsdecomposition). Interpretation: elements inside certain part areindistinguishable, while elements from different parts are distinguishable.There are incompatible events, e.g {{e1},{e2}} and {{e1,e2}}. This is logicalincompatibility analogical to the impossibility to have and simultaneously notto have the which-way information in the given experiment. The context is themaximal set of mutually compatible events. Each experiment has associated itscontext. In each context the extended probability is reduced to classicalprobability. Then the quadratic representation of events, partitions andprobability measures is developed. At the end the central concept of quadraticprobability spaces (which extend Kolmogorov probability spaces) is defined andstudied. In the next paper it will be shown that quantum mechanics can berepresented as the theory of Markov processes in the extended probabilitytheory (Einstein's vision of QM).
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