How Can Set-Valued Maps Be Integrated in Measurable Spaces?

In summary, if μ is nonatomic, then the upper integral of F with respect to μ is equal to the upper convex integral of F with respect to μ, which is equal to the integral of the upper convex hull of F with respect to μ. F is a set-valued map (or multifunction) from X to P(Y) without the empty set, and a measure μ is nonatomic if for any measurable set A with μ(A) > 0, there exists a measurable subset A1 of A such that μ(A) > μ(A1) > 0. The converse of this property is also true. For example, let F: X to P(Y) without the empty set be a measurable, integrably bounded set-val
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moh salem
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[tex]If \ \mu \ is\ nonatomic, \text{ }then \begin{equation*} \overline{\int_{X}Fd\mu }=\overline{co}\int_{X}F\text{ }d\mu =\int_{X}\overline{co}\text{ }F\text{ }d\mu , \end{equation*}[/tex][tex]Where, F :X\longrightarrow P(Y)\backslash \{\phi \}\ is \text{ } called\text{ }a\text{ } set-valued\text{ } map(or\text{ } multifunction.)[/tex] [tex]\text{ }And \text{ }a \text{ }measure\text{ } \mu \text{ }is \text{ }nonatomic \text{ }if \text{ }for \text{ }any \text{ }measurable \text{ }set \text{ }A \text{ }\text{ } with\text{ } \mu(A)>0 \text{ }\\there \text{ }exists \text{ }a \text{ }measurable \text{ }subset \text{ }A_{1} \text{ }of\text{ } A\text{ }such\text{ } that\text{ } \mu(A)>\mu (A_{1})>0.\\ Is \text{ }the \text{ }conversely \text{ }is \text{ }true \text{ }of \text{ }the \text{ }previous \text{ }property. \text{ }Please \text{ }help, \text{ }give\text{ }me \text{ }example.\\Thank \text{ }u.[/tex]
 
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  • #3
sorry, F : X→P(Y)\{φ} be a measurable, integrably bounded set-valued maps with closed images.
 

Related to How Can Set-Valued Maps Be Integrated in Measurable Spaces?

1. What is a set-valued map?

A set-valued map is a function that assigns a set of values to each point in a given domain. This means that for a given input, the output is not a single value but rather a set of values.

2. How is integration of set-valued map different from traditional integration?

Traditional integration deals with finding the area under a curve with a single-valued function as the integrand. Integration of set-valued map, on the other hand, involves finding the area under a curve with a set-valued function as the integrand. This requires a different approach and techniques.

3. What are some applications of integration of set-valued map?

Integration of set-valued map has various applications in fields such as optimization, economics, and game theory. It is also used in problems involving uncertainty and risk analysis.

4. What are some challenges in integrating set-valued maps?

One of the main challenges in integrating set-valued maps is the lack of a unified theory or approach. Different techniques and methods may need to be used depending on the specific set-valued function. Another challenge is the computational complexity involved in dealing with sets instead of single values.

5. How is the integration of set-valued maps related to measure theory?

Measure theory provides a mathematical framework for dealing with sets and their properties. Integration of set-valued maps relies heavily on the concepts and techniques of measure theory to define and compute the integration of set-valued functions.

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