In mathematics, lifting theory was first introduced by John von Neumann in a pioneering paper from 1931, in which he answered a question raised by Alfréd Haar. The theory was further developed by Dorothy Maharam (1958) and by Alexandra Ionescu Tulcea and Cassius Ionescu Tulcea (1961). Lifting theory was motivated to a large extent by its striking applications. Its development up to 1969 was described in a monograph of the Ionescu Tulceas. Lifting theory continued to develop since then, yielding new results and applications.

A lifting on a measure space ${\displaystyle (X,\Sigma ,\mu )}$ is a linear and multiplicative operator $${\displaystyle T:L^{\infty }(X,\Sigma ,\mu )\to {\mathcal {L}}^{\infty }(X,\Sigma ,\mu )}$$ which is a right inverse of the quotient map $${\displaystyle {\begin{cases}{\mathcal {L}}^{\infty }(X,\Sigma ,\mu )\to L^{\infty }(X,\Sigma ,\mu )\\f\mapsto [f]\end{cases}}}$$

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 ${\displaystyle r,s\in \mathbb {R} ,}$ $${\displaystyle T(r[f]+s[g])(p)=rT[f](p)+sT[g](p),}$$ $${\displaystyle T([f]\times [g])(p)=T[f](p)\times T[g](p).}$$

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. 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 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 $${\displaystyle T:L^{\infty }(X,\Sigma ,\mu )\to {\mathcal {L}}^{\infty }(X,\Sigma ,\mu )}$$ such that ${\displaystyle T\varphi =\varphi }$ on ${\displaystyle \operatorname {Supp} (\mu )}$ for all ${\displaystyle \varphi }$ in ${\displaystyle C_{b}(X,\tau ).}$ This is the same as requiring that ${\displaystyle TU\geq (U\cap \operatorname {Supp} (\mu ))}$ for all open sets ${\displaystyle U}$ in ${\displaystyle \tau .}$