the ${\sigma}$-algebras are assumed to be *complete*, i.e., to contain all the measure 0 sets.

An important feature of the Lebesgue measure spaces is

Let $p:(X,m)\to ({\overline}X,{\overline}m)$ be a *homomorphism* (*projection, factorization, quotient map*) of Lebesgue spaces, i.e., for any measurable set ${\overline}A\subset{\overline}X$ its preimage $A=p^{-1}({\overline}A)\subset X$ is also measurable, and $m(A)={\overline}m({\overline}A)$. Then the preimages $X_{{\overline}x}=p^{-1}({\overline}x),\, {\overline}x\in{\overline}X$, can be uniquely endowed with *conditional* probability measures $m_{{\overline}x}$ in such a way that $(X_{{\overline}x}, m_{{\overline}x})$ are Lebesgue spaces and the measure $m$ decomposes into an integral of the measures $m_{{\overline}x},\,{\overline}x\in{\overline}X,$ with respect to the *quotient measure* ${\overline}m$ on ${\overline}X$. Namely, for any function $f\in L^1(X,m)$ its restrictions $f_{{\overline}x}$ to $X_{{\overline}x}$ are measurable and belong to the respective spaces $L^1(X_{{\overline}x},m_{{\overline}x})$, the integrals ${\overline}f({\overline}x)= {\langle}f_{{\overline}x},m_{{\overline}x}{\rangle}$ depend on ${\overline}x$ measurably, and ${\langle}f,m{\rangle}= {\langle}{\overline}f, {\overline}m{\rangle}$ (cf. the classical Fubini theorem).

In fact, the above property follows from the classical Fubini theorem in view of Rokhlin’s

Any homomorphism $p:(X,m)\to ({\overline}X,{\overline}m)$ of Lebesgue spaces is uniquely (up to an isomorphism) determined by the signatures of the quotient measure ${\overline}m$ and of the conditional measures $m_{{\overline}x}$. Namely, let us denote by $I$ the unit interval endowed with the Lebesgue measure ${\lambda}$, and partition $I$ into a union of consecutive 