Multifunction Integration in Banach Spaces

Multifunction integration in Banach spaces addresses the challenge of integrating set-valued maps, or multifunctions, which assign to each point in a domain a non-empty subset of a Banach space rather than a single element. Central notions include strong integrals such as the Bochner integral, which requires measurability in the norm topology and integrable norm bounds, and weaker concepts like the Pettis integral, relying on scalar measurability and dual pairings. Extensions such as the Henstock–Kurzweil and McShane integrals accommodate cases beyond absolute convergence, offering finer control over integrability via gauge or partition criteria. Existence and measurability of selections—single-valued functions picked from the values of a multifunction—are guaranteed under compactness and convexity hypotheses. Decomposition theorems allow an integrable multifunction to be written as the sum of a well-behaved selection and a residual multifunction, clarifying structure and facilitating applications in differential inclusions, control theory, economics and image processing. Convergence theorems, including Vitali and Dominated Convergence analogues, ensure stability of the integral under limits of multifunction sequences or varying measures, establishing a robust analytical framework for multivalued phenomena.

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Recent advances have extended classical convergence principles to multifunction contexts under varying measures. A Vitali-type theorem now ensures that for sequences of scalar, vector and set-valued functions, weak or strong limits commute with Pettis and McShane integrals even when the underlying measure varies, broadening applicability to non-compact domains. Topological measure-space frameworks have been developed in which Pettis‐integrable multifunctions under vaguely convergent measures satisfy uniform convergence conditions; these results bridge functional analysis with probability and ergodic theory by controlling integrals in changing environments. Meanwhile, the existence of control measures for multimeasures in Banach spaces has been characterised: whenever the space avoids certain non-separable substructures, one can find a dominating scalar measure that governs the multimeasure, thereby importing scalar integration techniques to set-valued contexts and simplifying the study of measurability and integrability.

Multifunction Integration in Banach Spaces publication trend

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Technical terms

Multifunction: A mapping from a domain into the collection of non-empty subsets of a Banach space, generalising single-valued functions.

Pettis integral: A weak integral defined via duality, requiring each scalar evaluation against a continuous linear functional to be Lebesgue integrable.

Bochner integral: A strong integral for Banach-valued functions built from limits of simple functions, requiring strong measurability and integrability of the norm.

Henstock–Kurzweil integral: A gauge-based integral extending the Lebesgue integral to include non-absolute convergences, adaptable to Banach spaces.

Control measure: A scalar measure that dominates a multimeasure and allows reduction of set-valued integration problems to classical single-valued ones.

Decomposition property: A feature by which an integrable multifunction can be expressed as the sum of an integrable selection and a residual multifunction with stronger regularity.

References

1. Vitali Theorems for Varying Measures. Symmetry (2024).
2. Decompositions of Weakly Compact Valued Integrable Multifunctions. Mathematics (2020).
3. On control measures of multimeasures. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas (2022).
4. Convergence for varying measures in the topological case. Annali di Matematica Pura ed Applicata (1923 -) (2023).