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.[1] The theory was further developed by Dorothy Maharam (1958)[2] and by Alexandra Ionescu Tulcea and Cassius Ionescu Tulcea (1961).[3] 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.[4] Lifting theory continued to develop since then, yielding new results and applications.

Definitions

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A lifting on a measure space   is a linear and multiplicative operator   which is a right inverse of the quotient map  

where   is the seminormed Lp space of measurable functions and   is its usual normed quotient. In other words, a lifting picks from every equivalence class   of bounded measurable functions modulo negligible functions a representative— which is henceforth written   or   or simply   — in such a way that   and for all   and all      

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

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

Strong liftings

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Suppose   is complete and   is equipped with a completely regular Hausdorff topology   such that the union of any collection of negligible open sets is again negligible – this is the case if   is σ-finite or comes from a Radon measure. Then the support of     can be defined as the complement of the largest negligible open subset, and the collection   of bounded continuous functions belongs to  

A strong lifting for   is a lifting   such that   on   for all   in   This is the same as requiring that[7]   for all open sets   in  

Theorem. If   is σ-finite and complete and   has a countable basis then   admits a strong lifting.

Proof. Let   be a lifting for   and   a countable basis for   For any point   in the negligible set   let   be any character[8] on   that extends the character   of   Then for   in   and   in   define:     is the desired strong lifting.

Application: disintegration of a measure

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Suppose   and   are σ-finite measure spaces (  positive) and   is a measurable map. A disintegration of   along   with respect to   is a slew   of positive σ-additive measures on   such that

  1.   is carried by the fiber   of   over  , i.e.   and   for almost all  
  2. for every  -integrable function    in the sense that, for  -almost all   in     is  -integrable, the function   is  -integrable, and the displayed equality   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   is a Polish space[9] and   a separable Hausdorff space, both equipped with their Borel σ-algebras. Let   be a σ-finite Borel measure on   and   a  measurable map. Then there exists a σ-finite Borel measure   on   and a disintegration (*). If   is finite,   can be taken to be the pushforward[10]   and then the   are probabilities.

Proof. Because of the polish nature of   there is a sequence of compact subsets of   that are mutually disjoint, whose union has negligible complement, and on which   is continuous. This observation reduces the problem to the case that both   and   are compact and   is continuous, and   Complete   under   and fix a strong lifting   for   Given a bounded  -measurable function   let   denote its conditional expectation under   that is, the Radon-Nikodym derivative of[11]   with respect to   Then set, for every   in     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   and take the infimum over all positive   in   with   it becomes apparent that the support of   lies in the fiber over  

References

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  1. ^ von Neumann, John (1931). "Algebraische Repräsentanten der Funktionen "bis auf eine Menge vom Maße Null"". Journal für die reine und angewandte Mathematik (in German). 1931 (165): 109–115. doi:10.1515/crll.1931.165.109. MR 1581278.
  2. ^ Maharam, Dorothy (1958). "On a theorem of von Neumann". Proceedings of the American Mathematical Society. 9 (6): 987–994. doi:10.2307/2033342. JSTOR 2033342. MR 0105479.
  3. ^ Ionescu Tulcea, Alexandra; Ionescu Tulcea, Cassius (1961). "On the lifting property. I." Journal of Mathematical Analysis and Applications. 3 (3): 537–546. doi:10.1016/0022-247X(61)90075-0. MR 0150256.
  4. ^ Ionescu Tulcea, Alexandra; Ionescu Tulcea, Cassius (1969). Topics in the theory of lifting. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 48. New York: Springer-Verlag. MR 0276438. OCLC 851370324.
  5. ^ A subset   is locally negligible if it intersects every integrable set in   in a subset of a negligible set of     is complete if every locally negligible set is negligible and belongs to  
  6. ^ i.e., there exists a countable collection of integrable sets – sets of finite measure in   – that covers the underlying set  
  7. ^     are identified with their indicator functions.
  8. ^ A character on a unital algebra is a multiplicative linear functional with values in the coefficient field that maps the unit to 1.
  9. ^ A separable space is Polish if its topology comes from a complete metric. In the present situation it would be sufficient to require that   is Suslin, that is, is the continuous Hausdorff image of a Polish space.
  10. ^ The pushforward   of   under   also called the image of   under   and denoted   is the measure   on   defined by   for   in  .
  11. ^   is the measure that has density   with respect to