Given two probability measures on two probability spaces, ($\mu, X$) and ($\gamma, Y$), what's the sufficient and necessary condition such that there is a measurable mapping $f:X\rightarrow Y$, such that $f^\*\mu = \gamma$?



Image of probability measures under measurable mappings
There is a complete classification of probability spaces up to a measure-preserving
isomorphism.


Specifically, consider a category whose objects are triples
(X,Σ,μ), where X is a set, Σ is a σ-algebra
of measurable subsets on Σ,
and μ is a probability measure on (X,Σ).


If we want to have a nice description of morphisms
in terms of equivalence classes of point-set maps, we must
also require that (X,Σ,μ) is a compact probability space
in the sense of Marczewski's 1953 paper
“On compact measures”, but without countability assumptions.
We also assume completeness, since this does not change
isomorphism classes, but makes it easier to define morphisms.


Morphisms (X,Σ,μ)→(X',Σ',μ') are equivalence classes of maps
of sets f:X→X' such that f\*Σ'⊂Σ and μf\*=μ'.
The equivalence relation identifies f and g
if for all σ∈Σ' the symmetric difference f\*σ⊕g\*σ
has measure 0.


The better-known equivalence relation of equivalence almost
everywhere reduces to the above one if the involved spaces are countably
separated (like the real line), but in general one must use the above definition.


The isomorphism classes of objects in the resulting category
admit a complete classification.


First, any object canonically decomposes as a coproduct (disjoint union)
of ergodic spaces, i.e., spaces for which the automorphism
group has no nontrivial invariant subsets.


Secondly, the ergodic spaces admit a complete classification as follows.
First, the discrete spaces are ergodic.
Secondly, the nondiscrete ergodic spaces are isomorphic
to the infinite product of I copies of {0,1} (a discrete space),
where I is an infinite set, whose cardinality is the only invariant
of the resulting space.


The classification of isomorphism
classes (first stated for Boolean algebras Σ/N) is known as the von Neumann–Maharam theorem.
A 1965 theorem by Cassius Ionescu Tulcea (with subsequent
improvements by Vesterstrøm–Wils, Edgar, Graf, Fremlin, Rinkewitz) shows that isomorphisms classes of such
Boolean algebras coincide with appropriately defined
isomorphism classes of probability spaces.


(This claim also holds if we remove the requirement that (X,Σ,μ) is compact,
in which case we must first localize the category at all morphisms f
such that the induced map of Boolean algebras Σ'/N'→Σ/N is
an isomorphism. Completeness can also be removed if the notion
of measurability is adjusted accordingly: elements in f\*Σ' must be symmetric differences of elements in Σ and a subset of a measure 0 set.)


Thus, the original question can be answered as follows:
there is a measurable mapping f:X→Y such that f\*μ=γ
if and only if the decompositions of X and Y into their ergodic
components have the same measures for each type of ergodic summand,
and the discrete parts are isomorphic.


