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.
Definitions A lifting on a measure space ( X , Σ , μ ) {\displaystyle (X,\Sigma ,\mu )} is a linear and multiplicative operator
T : L ∞ ( X , Σ , μ ) → L ∞ ( X , Σ , μ ) {\displaystyle T:L^{\infty }(X,\Sigma ,\mu )\to {\mathcal {L}}^{\infty }(X,\Sigma ,\mu )}
which is a right inverse of the quotient map
{ L ∞ ( X , Σ , μ ) → L ∞ ( X , Σ , μ ) f ↦ [ f ] {\displaystyle {\begin{cases}{\mathcal {L}}^{\infty }(X,\Sigma ,\mu )\to L^{\infty }(X,\Sigma ,\mu )\\f\mapsto [f]\end{cases}}}
where L ∞ ( X , Σ , μ ) {\displaystyle {\mathcal {L}}^{\infty }(X,\Sigma ,\mu )} is the seminormed Lp space of measurable functions and L ∞ ( X , Σ , μ ) {\displaystyle L^{\infty }(X,\Sigma ,\mu )} is its usual normed quotient. In other words, a lifting picks from every equivalence class [ f ] {\displaystyle [f]} of bounded measurable functions modulo negligible functions a representative— which is henceforth written T ( [ f ] ) {\displaystyle T([f])} or T [ f ] {\displaystyle T[f]} or simply T f {\displaystyle Tf} — in such a way that T [ 1 ] = 1 {\displaystyle T[1]=1} and for all p ∈ X {\displaystyle p\in X} and all r , s ∈ R , {\displaystyle r,s\in \mathbb {R} ,}
T ( r [ f ] + s [ g ] ) ( p ) = r T [ f ] ( p ) + s T [ g ] ( p ) , {\displaystyle T(r[f]+s[g])(p)=rT[f](p)+sT[g](p),}
T ( [ f ] × [ g ] ) ( p ) = T [ f ] ( p ) × T [ 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.
Existence of liftings Theorem. Suppose ( X , Σ , μ ) {\displaystyle (X,\Sigma ,\mu )} is complete. Then ( X , Σ , μ ) {\displaystyle (X,\Sigma ,\mu )} admits a lifting if and only if there exists a collection of mutually disjoint integrable sets in Σ {\displaystyle \Sigma } whose union is X . {\displaystyle X.}
In particular, if ( X , Σ , μ ) {\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 ( X , Σ , μ ) {\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.
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