Integral of a correspondence

In mathematics, the integral of a correspondence is a generalization of the integration of single-valued functions to correspondences (i.e., set-valued functions).

The first notion of the integral of a correspondence is due to Aumann in 1965,^[1] with a different approach by Debreu appearing in 1967.^[2] Integrals of correspondences have applications in general equilibrium theory in mathematical economics,^[3][4] random sets in probability theory,^[5][6] partial identification in econometrics,^[7] and fuzzy numbers in fuzzy set theory.^[8]

Preliminaries

Correspondences

A correspondence ${\displaystyle \varphi :X\rightrightarrows Y}$ is a function ${\displaystyle \varphi :X\rightarrow {\mathcal {P}}(Y)}$, where ${\displaystyle {\mathcal {P}}(Y)}$ is the power set of ${\displaystyle Y}$. That is, ${\displaystyle \varphi }$ assigns each point ${\displaystyle x\in X}$ with a set ${\displaystyle \varphi (x)\subset Y}$.

Selections

A selection ${\displaystyle f}$ of a correspondence ${\displaystyle \varphi :X\rightrightarrows Y}$ is a function ${\displaystyle f:X\rightarrow Y}$ such that ${\displaystyle f(x)\in \varphi (x)}$ for every ${\displaystyle x\in X}$.

If ${\displaystyle X}$ can be seen as a measure space ${\displaystyle (X,{\mathcal {X}},\mu )}$ and ${\displaystyle Y}$ as a Banach space ${\displaystyle (Y,||\cdot ||)}$, then one can define a measurable selection ${\displaystyle f}$ as an ${\displaystyle {\mathcal {X}}}$-measurable function^{[nb 1]} ${\displaystyle f}$ such that ${\displaystyle f(x)\in \varphi (x)}$ for μ-almost all ${\displaystyle x\in X}$.^{[5][nb 2]}

Definitions

The Aumann integral

Let ${\displaystyle (X,{\mathcal {X}},\mu )}$ be a measure space and ${\displaystyle (Y,||\cdot ||)}$ a Banach space. If ${\displaystyle \varphi :X\rightrightarrows Y}$ is a correspondence, then the Aumann integral of ${\displaystyle \varphi }$ is defined as

${\displaystyle \int _{X}\varphi d\mu$ :=\left\{\int _{X}fd\mu :f{\text{ is a measurable selection of }}\varphi \right\}}

where the integrals ${\displaystyle \int _{X}fd\mu }$ are Bochner integrals.

Example: let the underlying measure space be ${\displaystyle ([0,1],{\mathcal {L}},\lambda )}$, and a correspondence ${\displaystyle \varphi$ :[0,1]\rightrightarrows \mathbb {R} } be defined as ${\displaystyle \varphi (x)=\{2,3\}}$ for all ${\displaystyle x\in [0,1]}$. Then the Aumman integral of ${\displaystyle \varphi }$ is ${\displaystyle \int _{X}\varphi d\lambda =[2,3]}$.

The Debreu integral

Debreu's approach to the integration of a correspondence is more restrictive and cumbersome, but directly yields extensions of usual theorems from the integration theory of functions to the integration of correspondences, such as Lebesgue's Dominated convergence theorem.^[3] It uses Rådström's embedding theorem to identify convex and compact valued correspondences with subsets of a real Banach space, over which Bochner integration is straightforward.^[2]

Let ${\displaystyle (X,{\mathcal {X}},\mu )}$ be a measure space, ${\displaystyle (Y,||\cdot ||)}$ a Banach space, and ${\displaystyle {\mathcal {K}}\subset {\mathcal {P}}(Y)}$ the set of all its convex and compact subsets. Let ${\displaystyle \varphi :X\to {\mathcal {K}}}$ be a convex and compact valued correspondence from ${\displaystyle X}$ to ${\displaystyle Y}$. By Rådström's embedding theorem, ${\displaystyle {\mathcal {K}}}$ can be isometrically embedded as a convex cone ${\displaystyle C}$ in a real Banach space ${\displaystyle ({\mathcal {Y}},||\cdot ||_{\mathcal {Y}})}$, in such a way that addition and multiplication by nonnegative real numbers in ${\displaystyle {\mathcal {Y}}}$ induces the corresponding operation in ${\displaystyle {\mathcal {K}}}$.

Let ${\displaystyle \varphi ^{*}:X\rightrightarrows C}$ be the "image" of ${\displaystyle \varphi }$ under the embedding defined above, in the sense that ${\displaystyle \varphi ^{*}(x)\subset C}$ is the image of ${\displaystyle \varphi (x)}$ under this embedding for every ${\displaystyle x\in X}$. For each pair of ${\displaystyle {\mathcal {X}}}$-simple functions ${\displaystyle f,g:X\to C}$, define the metric ${\displaystyle \Delta (f,g)=\int _{X}||f-g||_{\mathcal {Y}}d\mu }$.

Then we say that ${\displaystyle \varphi }$ is integrable if ${\displaystyle \varphi ^{*}}$ is integrable in the following sense: there exists a sequence of ${\displaystyle {\mathcal {X}}}$-simple functions ${\displaystyle (f_{n})_{n}}$ from ${\displaystyle X}$ to ${\displaystyle C}$ which are Cauchy in the metric ${\displaystyle \Delta }$ and converge in measure to ${\displaystyle \varphi ^{*}}$. In this case, we define the integral of ${\displaystyle \varphi ^{*}}$ to be

${\displaystyle \int _{X}\varphi ^{*}d\mu$ :=\lim _{n\to \infty }\int _{X}f_{n}d\mu \in C}

where the integrals ${\displaystyle \int _{X}f_{n}d\mu }$ are again simply Bochner integrals in the space ${\displaystyle ({\mathcal {Y}},||\cdot ||_{\mathcal {Y}})}$, and the result still belongs ${\displaystyle C}$ since it is a convex cone. We then uniquely identify the Debreu integral of ${\displaystyle \varphi }$ as^[5]

${\displaystyle \int _{X}\varphi d\mu$ :=E\in {\mathcal {K}}}

such that ${\displaystyle E^{*}=\int _{X}\varphi ^{*}d\mu \in C}$. Since every embedding is injective and surjective onto its image, the Debreu integral is unique and well-defined.

Notes

1. Measurable in the sense of Bochner measurable: there exists a sequence ${\displaystyle (\psi _{n})_{n}}$ of simple functions from ${\displaystyle X}$ to ${\displaystyle Y}$ such that ${\displaystyle \lim _{n\to \infty }||f(x)-\psi _{n}(x)||=0}$ for μ-almost all ${\displaystyle x\in X}$.
2. A stronger definition sometimes used requires ${\displaystyle f}$ to be measurable and ${\displaystyle f(x)\in \varphi (x)}$ for all ${\displaystyle x\in X}$. ^[4]