Product Vector Liftings

• Product vector liftings are linear operators on function spaces over product probability spaces that select canonical representatives, maintaining consistency with tensor structures.
• They generalize classical L∞ liftings to all L^p spaces, enabling measurable selections and modifications crucial for stochastic process theory.
• Their construction leverages Fubini compatibility and basis-adapted techniques to address nonseparable contexts and extend measure-theoretic results.

A product vector lifting is a linear operator on function spaces over product probability spaces that selects canonical representatives from equivalence classes defined by almost-everywhere equality and interacts compatibly with the tensor product structure. These liftings generalize the classical concept of liftings on $L^\infty$ to all $L^p$ spaces and play a central role in measurable selection, stochastic process theory, and the algebraic structure of product $L^p$ spaces. Product vector liftings enable rigorous constructions and modifications in stochastic analysis without measure-completeness assumptions and elucidate structure in nonseparable probability-theoretic contexts (Burke et al., 27 Nov 2025).

1. Foundation: Vector Liftings on $L^p$ Spaces

Let $(X,\Sigma,\mu)$ be a probability space. For $0 < p \le \infty$, define $L^p(\mu)$ as the space of $\mu$-measurable functions (modulo almost-everywhere equality) with the $L^p$ norm finite. A vector lifting is a linear primitive lifting $\rho: L^p(\mu) \to L^p(\mu)$ that selects for each equivalence class a representative such that for $f=_\mu g$ (i.e., $f=g$ $\mu$-almost everywhere), $\rho(f) = \rho(g)$. The operator $\rho$ is required to be linear and to fix constants, and for all $f$, $\rho(f) = f$ $\mu$-almost everywhere.

A key structural property is that vector liftings always exist for $L^p(\mu)$ when $p \in [0,\infty]$ and $\mu(X)>0$ (Burke et al., 27 Nov 2025). This is achieved by choosing a linear complement to the null subspace. However, for $p<\infty$, positive linear liftings do not exist unless $\mu$ is purely atomic.

2. Product Structures and Fubini Properties

For two probability spaces $(X,\mathfrak S, \mu)$ and $(Y, \mathfrak T, \nu)$, the $L^p$-structure on $X \times Y$ is governed by the completed product measure $\upsilon = \mu \otimes \nu$. The Fubini property $[C]$ (and its symmetric form $[{C}]$) ensures that integrals and measurability on the product are compatible with those on the marginals. Preservation of $[C]$ is crucial when constructing product vector liftings and when extending measures by null sets or ideals beyond the original $\sigma$-algebra.

Construction of product vector liftings typically requires care to ensure that liftings on the factors extend correctly to the product, particularly for infinite-dimensional $L^p$ spaces and nonseparable situations.

3. Existence and Explicit Construction of Product Vector Liftings

Given vector liftings $\gamma$ on $L^p(\mu)$ and $\eta$ on $L^p(\nu)$, a product vector lifting $\pi$ on $L^p(\mu \otimes \nu)$ is a linear primitive lifting that satisfies

$\pi(f \otimes g) = \gamma(f) \otimes \eta(g)$

for all $f \in L^p(\mu)$ and $g \in L^p(\nu)$. The main existence theorem is as follows (Burke et al., 27 Nov 2025):

Theorem (Product Vector Lifting):

Let both marginals be Radon probability spaces, each with a strong vector lifting, and suppose the product measure has the Fubini property $[C]$ and $[{C}]$. Then there exists a strong vector lifting $\pi$ on the product such that $\pi(f \otimes g) = \gamma(f) \otimes \eta(g)$, and the horizontal and vertical sections of $\pi(h)$ for $h \in L^p(\mu \otimes \nu)$ again belong to the correct $L^p$-spaces.

The construction proceeds by choosing bases for $L^p(\mu)$ and $L^p(\nu)$ adapted so that simple tensors of continuous functions appear first, extending to a basis of the product space, and defining $\pi$ to fix this basis. Fubini's theorem guarantees that sections remain in $L^p$.

4. Measure Extensions and Null Ideals

Measure-space completions and extensions play a critical role in the definition and application of product vector liftings. Given a measure space $(X, \Sigma, \mu)$ and a family $F$ of null sets, one may extend $\mu$ to a larger $\sigma$-algebra so that every set in the extension is equivalent modulo a null set to a set in $\Sigma$. For products, the key example is the right-nil null ideal

$$\mathfrak N = \{ E \subset X \times Y : \exists N \subset X,\ \mu(N)=0,\ \forall x \notin N,\ E_x \in \mathcal N(\nu) \}$$

where $E_x = \{ y \in Y : (x, y) \in E \}$.

Preserving the Fubini property $[C]$ and ensuring the existence of product vector liftings relies on the ability to extend measures to handle such null ideals while maintaining measurability constraints (Burke et al., 27 Nov 2025).

5. Applications to Stochastic Processes and Measurable Modifications

Product vector liftings yield a powerful machinery for constructing and characterizing measurable modifications of stochastic processes. For a family of measurable maps $Q_x: Y \to \Gamma$ indexed by $x \in X$, a measurable modification is a map $U: X \times Y \to \Gamma$ such that $U(x, \cdot) = Q_x$ modulo null sets.

The main result [(Burke et al., 27 Nov 2025), Thm. 5.10] is that, under $[C]$ and the rectangular measurability hypothesis, the following are equivalent:

• The process admits an $L^p$-modification,
• The graphical map $(x,y) \mapsto Q(x,y)$ is measurable,
• $Q$ is measurable outside a set in the right-nil null ideal,
• There exists a single measurable $U$ that agrees almost everywhere in both variables.

When product vector liftings exist, one can exhibit a canonical measurable modification of a process by explicit algebraic construction, bypassing separability assumptions or maximal-inequality arguments typical in classical stochastic analysis.

6. Algebraic and Analytic Implications

Product vector liftings extend the range of functional-analytic techniques available for product $L^p$ spaces. They enable precise representative selection in algebraic and probabilistic frameworks, crucial for constructions in descriptive set theory, integration theory without completeness, and for rigorous handling of tensor-product structures in infinite-dimensional settings.

These liftings underpin the existence (and sometimes uniqueness) of modifications in stochastic process theory and facilitate the extension of measure-theoretic results (such as measurable selection theorems) to settings lacking classical completeness or separability.

7. Summary Table: Key Properties of Product Vector Liftings

Property	Details	Source
Linearity	Both the marginal and product liftings are linear on $L^p$	(Burke et al., 27 Nov 2025)
Basis adaptation	Construction relies on extending adapted bases from marginals to product	(Burke et al., 27 Nov 2025)
Fubini compatibility	Existence predicated on Fubini property preservation under extension	(Burke et al., 27 Nov 2025)
Existence for all $p \in [0,\infty]$	Product vector liftings extend to all $L^p$, in contrast to classical liftings which are $L^\infty$ only	(Burke et al., 27 Nov 2025)
Measurable process modifications	Directly characterize and construct measurable modifications from product liftings	(Burke et al., 27 Nov 2025)

Product vector liftings thus serve as a unifying structure in probability, functional analysis, and stochastic process theory, enabling explicit, algebraic, and basis-dependent constructions that sidestep classical completeness requirements and facilitate extensions across product spaces.