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.
Probability – Random Variables with Same Distribution
measure-theorypr.probability
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Let us consider more closely the question about space-filling curves. The Peano curve and the Hilbert curve, and several other variations of them, have parametrizations $[0,1]\to[0,1]^2$ that actually take the 1-dimensional Lebesgue measure on $[0,1]\, ,$ $\mathcal{L} _1\, ,$ to the 2-dimensional Lebesgue measure on $[0,1]^2\, ,$ $\mathcal{L} _2\, ,$ by push-forward. Notably, the Peano curve $\gamma: [0,1]\to[0,1]^2$ is originally defined in terms of ternary expansions of real numbers, which makes it particularly simple to check the independence of the coordinates. The Hilbert curve also admits a (slightly less) simple description in terms of binary expansions. Here are the details; I'll try to give you an abstract-nonsense-friendly description.
Let's start by some well known facts. The map that takes a sequence $a:=(a_1,a_2,\dots)\in\ 3^\mathbb{N}$ (here $3:=\{0,1,2\}$ and $\mathbb{N}$ is the set of positive integers) into its value as ternary expansion, namely $\mathrm{v}(a):= \sum_ {n\ge1} 3^{-n}a_ n\in[0,1]$, is a continuous surjective map (and bijective, up to removing a certain countable, null subset $D$ of the domain) :
$$\operatorname{v}:3^\mathbb{N}\to[0,1]\, .$$
Also, this map takes the measure $\mathbf{m}$, product of countably many copies of the uniform probability measure on $3$ to the measure $\mathcal{L} _1$ ("the base 3 digits of a real number are independent and uniformly distributed". Any other base of course works as well).
Now, at the level of the ternary sequences we have the nice and simple Cantor bijection $3^\mathbb{N}\to 3^\mathbb{N}\times 3^\mathbb{N}$
"split the sequence of digits into the sequence of odd-position digits and the sequence of even-position digits"
$$C:3^\mathbb{N}\ni (a_n)\mapsto \big(\, (a_{2n-1}), (a_{2n})\, \big) \in 3^\mathbb{N}\times 3^\mathbb{N} $$
which is easily seen to be a compact metric space homeomorphism that takes the measure $\mathbf{m}$ into the product measure $\mathbf{m}\otimes\mathbf{m}$. Note that this homeomorphism does not pass through the quotient map $\mathrm{v}$, for in general sequences with the same value do not produce sequences with the same value by extraction of a subsequence. However, removing the above mentioned countable set $D$ the map $\mathbf{v}$ becomes bijective and you do have correspondingly a bi-measurable, a.e. defined (or everywhere but non-continuous) map of the unit interval to the unit square that takes the measure $\mathcal{L} _1$ to the measure $\mathcal{L} _2$. All that is quite standard. Now your question cames quite naturally, as a request for a commutative square:
Can we find another measure preserving homeomorphism $\Gamma:3^\mathbb{N}\to > 3^\mathbb{N}\times 3^\mathbb{N}$ that induces a map $\gamma: > [0,1]\to[0,1]\times[0,1]$ through the map $\mathbf{v}$, that is, such that $\gamma\circ\mathbf{v}=(\mathbf{v}\times\mathbf{v})\circ\Gamma$ ?
Since $\mathbf{v}$ is a quotient map, this map $\gamma$ will be automatically a continuous surjection, that also takes $\mathcal{L} _1$ to $\mathcal{L} _2$. The answer is yes, and this is Peano's construction (he was not interested in the measure-theoretic property, but this also follows immediately from the definition). It's the way he constructed his example in the celebrated paper dated 1890 on Mathematische Annalen, " Sur une courbe, qui remplit toute une aire plane ". Here's Peano definition of the map $\Gamma$: extract as before the sequence of the odd-position digits and the even-position digits, but first invert every odd-position digit whenever there are an odd number of odd even-position digits before it, and invert every even-position digit whenever there are an odd number of odd odd-position digits before it. Here "invert" just means taking $x\in3$ to $2-x$, that is $0$ to $2$, $1$ to $1$, $2$ to $0$. Translating this definition in a formula, is not difficult to check it defines a homeomorphism of the form $\Gamma=C \circ \phi$, compatible with the map $\mathbf{v}$. The measure property is quite obvious, since the "inverting digit map" $ \phi:3^\mathbb{N}\to 3^\mathbb{N}$ is clearly an involutory preserving measure homeomorphism. For the Hilbert curve, the digit description has to be done in terms of binary representation, and it is slightly less simple (I have it written somewhere and will look for it and quote here at request) but everything works as well.
Let me finish with an historical note. Of course, what is not easy in the short Peano's paper is to understand what's going on geometrically. He made no picture in this paper, although the graphical iterative construction was perfectly clear to him, and was with all probability his starting point —he made an ornamental tiling showing a picture of the curve in his home in Turin. His choice to avoid any appeal to graphical visualization was no doubt motivated by a desire for a well-founded, completely rigorous proof owing nothing to pictures, in the spirit of the program of arithmetization of analysis. In the conclusion of his paper he observed incidentally that the same construction may be made with all odd basis, and even basis too, although in the latter case, by slightly more complicated formulae (hence less elegant from his viewpoint). In order to make Peano's example more accessible to the mathematical community, a couple of years later Hilbert wrote on the same journal the very clear geometric construction that we know. He chose the Peano construction in base 2 because it is simpler from the graphical point of view.
This is a fascinating topic. One impressive systematic study of symmetries is in the book by Olav Kallenberg (2005)
In there, though, the measurable space has to have some structure to get the most out of the results.
I don't know of any systematic applications of Lie groups to probability theory. However, there are here and there some interesting results. For instance, this book contains a study of measures that are invariant under O(n).
There is also plenty of results and applications of discrete symmetries (among others) in here:
Maybe one should ask a community wiki question where everyone tries to list the results they know. That would be a very interesting list!
Edit: I recently came across this book that is a quite relevant reference for studying symmetries of probability measures:
It has an extensive discussion on Lie groups ans Lie algebras.
Edit 2: Another book with an extensive discussion on Lie groups in Probability and Statistics!
Best Answer
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 (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.