A lifting on a measure space ${\displaystyle (X,\Sigma ,\mu )}$  is a linear and multiplicative operator

where ${\displaystyle {\mathcal {L}}^{\infty }(X,\Sigma ,\mu )}$  is the seminormed L^p space of measurable functions and ${\displaystyle L^{\infty }(X,\Sigma ,\mu )}$  is its usual normed quotient. In other words, a lifting picks from every equivalence class ${\displaystyle [f]}$  of bounded measurable functions modulo negligible functions a representative— which is henceforth written ${\displaystyle T([f])}$  or ${\displaystyle T[f]}$  or simply ${\displaystyle Tf}$  — in such a way that ${\displaystyle T[1]=1}$  and for all ${\displaystyle p\in X}$  and all ${\displaystyle r,s\in \mathbb {R} ,}$ 

Liftings are used to produce disintegrations of measures, for instance conditional probability distributions given continuous random variables, and fibrations of Lebesgue measure on the level sets of a function.

Theorem. Suppose ${\displaystyle (X,\Sigma ,\mu )}$  is complete.^[5] Then ${\displaystyle (X,\Sigma ,\mu )}$  admits a lifting if and only if there exists a collection of mutually disjoint integrable sets in ${\displaystyle \Sigma }$  whose union is ${\displaystyle X.}$  In particular, if ${\displaystyle (X,\Sigma ,\mu )}$  is the completion of a σ-finite^[6] measure or of an inner regular Borel measure on a locally compact space, then ${\displaystyle (X,\Sigma ,\mu )}$  admits a lifting.

The proof consists in extending a lifting to ever larger sub-σ-algebras, applying Doob's martingale convergence theorem if one encounters a countable chain in the process.

Suppose ${\displaystyle (X,\Sigma ,\mu )}$  is complete and ${\displaystyle X}$  is equipped with a completely regular Hausdorff topology ${\displaystyle \tau \subseteq \Sigma }$  such that the union of any collection of negligible open sets is again negligible – this is the case if ${\displaystyle (X,\Sigma ,\mu )}$  is σ-finite or comes from a Radon measure. Then the support of ${\displaystyle \mu ,}$  ${\displaystyle \operatorname {Supp} (\mu ),}$  can be defined as the complement of the largest negligible open subset, and the collection ${\displaystyle C_{b}(X,\tau )}$  of bounded continuous functions belongs to ${\displaystyle {\mathcal {L}}^{\infty }(X,\Sigma ,\mu ).}$ 

A strong lifting for ${\displaystyle (X,\Sigma ,\mu )}$  is a lifting

Theorem. If ${\displaystyle (\Sigma ,\mu )}$  is σ-finite and complete and ${\displaystyle \tau }$  has a countable basis then ${\displaystyle (X,\Sigma ,\mu )}$  admits a strong lifting.

Proof. Let ${\displaystyle T_{0}}$  be a lifting for ${\displaystyle (X,\Sigma ,\mu )}$  and ${\displaystyle U_{1},U_{2},\ldots }$  a countable basis for ${\displaystyle \tau .}$  For any point ${\displaystyle p}$  in the negligible set

Suppose ${\displaystyle (X,\Sigma ,\mu )}$  and ${\displaystyle (Y,\Phi ,\nu )}$  are σ-finite measure spaces (${\displaystyle \mu ,\mu }$  positive) and ${\displaystyle \pi :X\to Y}$  is a measurable map. A disintegration of ${\displaystyle \mu }$  along ${\displaystyle \pi }$  with respect to ${\displaystyle \nu }$  is a slew ${\displaystyle Y\ni y\mapsto \lambda _{y}}$  of positive σ-additive measures on ${\displaystyle (\Sigma ,\mu )}$  such that

1. ${\displaystyle \lambda _{y}}$  is carried by the fiber ${\displaystyle \pi ^{-1}(\{y\})}$  of ${\displaystyle \pi }$  over ${\displaystyle y}$ , i.e. ${\displaystyle \{y\}\in \Phi }$  and ${\displaystyle \lambda _{y}\left((X\setminus \pi ^{-1}(\{y\})\right)=0}$  for almost all ${\displaystyle y\in Y}$ 
2. for every ${\displaystyle \mu }$ -integrable function ${\displaystyle f,}$ 
   $${\displaystyle \int _{X}f(p)\;\mu (dp)=\int _{Y}\left(\int _{\pi ^{-1}(\{y\})}f(p)\,\lambda _{y}(dp)\right)\nu (dy)\qquad (*)}$$

    
   in the sense that, for ${\displaystyle \nu }$ -almost all ${\displaystyle y}$  in ${\displaystyle Y,}$  ${\displaystyle f}$  is ${\displaystyle \lambda _{y}}$ -integrable, the function
   $${\displaystyle y\mapsto \int _{\pi ^{-1}(\{y\})}f(p)\,\lambda _{y}(dp)}$$

    
   is ${\displaystyle \nu }$ -integrable, and the displayed equality ${\displaystyle (*)}$  holds.

Disintegrations exist in various circumstances, the proofs varying but almost all using strong liftings. Here is a rather general result. Its short proof gives the general flavor.

Theorem. Suppose ${\displaystyle X}$  is a Polish space^[9] and ${\displaystyle Y}$  a separable Hausdorff space, both equipped with their Borel σ-algebras. Let ${\displaystyle \mu }$  be a σ-finite Borel measure on ${\displaystyle X}$  and ${\displaystyle \pi :X\to Y}$  a ${\displaystyle \Sigma ,\Phi -}$ measurable map. Then there exists a σ-finite Borel measure ${\displaystyle \nu }$  on ${\displaystyle Y}$  and a disintegration (*). If ${\displaystyle \mu }$  is finite, ${\displaystyle \nu }$  can be taken to be the pushforward^[10] ${\displaystyle \pi _{*}\mu ,}$  and then the ${\displaystyle \lambda _{y}}$  are probabilities.

Proof. Because of the polish nature of ${\displaystyle X}$  there is a sequence of compact subsets of ${\displaystyle X}$  that are mutually disjoint, whose union has negligible complement, and on which ${\displaystyle \pi }$  is continuous. This observation reduces the problem to the case that both ${\displaystyle X}$  and ${\displaystyle Y}$  are compact and ${\displaystyle \pi }$  is continuous, and ${\displaystyle \nu =\pi _{*}\mu .}$  Complete ${\displaystyle \Phi }$  under ${\displaystyle \nu }$  and fix a strong lifting ${\displaystyle T}$  for ${\displaystyle (Y,\Phi ,\nu ).}$  Given a bounded ${\displaystyle \mu }$ -measurable function ${\displaystyle f,}$  let ${\displaystyle \lfloor f\rfloor }$  denote its conditional expectation under ${\displaystyle \pi ,}$  that is, the Radon-Nikodym derivative of^[11] ${\displaystyle \pi _{*}(f\mu )}$  with respect to ${\displaystyle \pi _{*}\mu .}$  Then set, for every ${\displaystyle y}$  in ${\displaystyle Y,}$  ${\displaystyle \lambda _{y}(f):=T(\lfloor f\rfloor )(y).}$  To show that this defines a disintegration is a matter of bookkeeping and a suitable Fubini theorem. To see how the strongness of the lifting enters, note that