mathematics

Integral of a correspondence

Integral of a correspondence is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Integral of a correspondence rather than just read about it. In short: 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, with a different approach by Debreu appearing in 1967.

Key takeaways

Integral of a correspondence belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Integral of a correspondence to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Integral of a correspondence from memory before moving on to harder problems.

Reference excerpt

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, with a different approach by Debreu appearing in 1967. Integrals of correspondences have applications in general equilibrium theory in mathematical economics, random sets in probability theory, partial identification in econometrics, and fuzzy numbers in fuzzy set theory.

Preliminaries

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

Selections A selection f {\displaystyle f} of a correspondence φ : X ⇉ Y {\displaystyle \varphi :X\rightrightarrows Y} is a function f : X → Y {\displaystyle f:X\rightarrow Y} such that f ( x ) ∈ φ ( x ) {\displaystyle f(x)\in \varphi (x)} for every x ∈ X {\displaystyle x\in X} . If X {\displaystyle X} can be seen as a measure space ( X , X , μ ) {\displaystyle (X,{\mathcal {X}},\mu )} and Y {\displaystyle Y} as a Banach space ( Y , | | ⋅ | | ) {\displaystyle (Y,||\cdot ||)} , then one can define a measurable selection f {\displaystyle f} as an X {\displaystyle {\mathcal {X}}} -measurable function f {\displaystyle f} such that f ( x ) ∈ φ ( x ) {\displaystyle f(x)\in \varphi (x)} for μ-almost all x ∈ X {\displaystyle x\in X} .

Definitions

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

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

where the integrals ∫ X f d μ {\displaystyle \int _{X}fd\mu } are Bochner integrals. Example: let the underlying measure space be ( [ 0 , 1 ] , L , λ ) {\displaystyle ([0,1],{\mathcal {L}},\lambda )} , and a correspondence φ : [ 0 , 1 ] ⇉ R {\displaystyle \varphi :[0,1]\rightrightarrows \mathbb {R} } be defined as φ ( x ) = { 2 , 3 } {\displaystyle \varphi (x)=\{2,3\}} for all x ∈ [ 0 , 1 ] {\displaystyle x\in [0,1]} . Then the Aumman integral of φ {\displaystyle \varphi } is ∫ X φ d λ = [ 2 , 3 ] {\displaystyle \int _{X}\varphi d\lambda =[2,3]} .

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Integral of a correspondence

Start with the simplest possible case. Write down what Integral of a correspondence claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Integral of a correspondence before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Integral of a correspondence ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Integral of a correspondence

In research: Integral of a correspondence appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Integral of a correspondence in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Integral of a correspondence is common in secondary-school and first-year university syllabi. It links to neighbouring topics Functional analysis, Mathematical economics, so understanding it makes those chapters shorter.
In everyday life: Look for Integral of a correspondence outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Integral of a correspondence in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Integral of a correspondence means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Integral of a correspondence out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Integral of a correspondence in simple terms?

Why does Integral of a correspondence matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Integral of a correspondence?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Integral of a correspondence.