Consider probability space W with pair of random variables having same distribution. On how much this variables distinct in terms of W symmetries? Namely, let's talk about automorphism as measure-preserving self-mapping of W defined almost everywhere. The following question must be well studied. When such random variables may be combined by some automorphism of W (up to measure-zero set, of course)? Sorry for bad english.



Random variables with same distribution
There are some obvious restrictions in the case when the base probability
space $W$ is allowed to have atoms. For instance, if $W$ consists of an atom
and a continuous part with equal masses $1/2$, then their indicator functions
have the same distribution, but an automorphism in question clearly does not
exist.


A slightly more involved counterexample is provided by the following pair of
random variables defined on the unit interval endowed with the usual Lebesgue
probability measure. One is the identity map, and the other one is $x\mapsto
2x$ (mod 1).


I will give a complete answer to your question in the situation when the base
probability space $(W,P)$ is a [Lebesgue
(standard) probability space](http://en.wikipedia.org/wiki/Standard_probability_space) (which is the only reasonable generality in
probability nowadays, although there is a lot of people here who are very fond
of discussing various exotic if not outright pathological measure spaces). The
*signature* of a measure space is the mass of its non-atomic part plus
the non-increasing sequence of the weights of its atoms. Rohlin (see the above
Wikipedia article for a reference to his 1949 article very appropriately called "On
the fundamental ideas of measure theory") proved that, up to isomorphism,
Lebesgue spaces are completely characterized by their signatures. In
particular, there is only one purely non-atomic Lebesgue measure space, which
is just the unit interval with the Lebesgue measure on it (whence the term).


It is less known that Rohlin also obtained a complete classification of
homomorphisms of Lebesgue spaces (equivalently, of their measurable
partitions, or of complete sub-$\sigma$-algebras). Signature of the quotient
measure and signatures of the conditional measures associated with the
homomorphism provide an obvious system of conjugacy invariants of such
homomorphisms. Rohlin proved that this system is, in fact, a complete system
of invariants. In the simplest purely non-atomic case it means that any
homomorphism of Lebesgue spaces with a purely non-atomic quotient space and
purely non-atomic conditional measures is conjugate to the coordinate
projection of the unit square onto the unit interval (both being endowed with
the canonical Lebesgue measures).


Applied to your original question, Rohlin's classification implies that two
random variables with the same distribution on a Lebesgue space are equivalent
in your sense if and only if for a.e. value taken by these variables the
corresponding conditional measure spaces are isomorphic, i.e., have the same
signature. In particular, the latter is the case if almost all conditional
measures are purely non-atomic.


