Measurability and integration of set-valued maps

[moh salem]

What is the difference between the measurable set-valued maps and measurable single-valued map?
What is the difference between the integrable set-valued maps and integrable single-valued map?
With illustrative examples, if possible?
Thank you very.

 

[mathman]

I am not sure what you are trying to get at. A set value map is a function of sets while a single value map is a function of points.

 

[moh salem]

I mean from the first paragraph is to:

$$Let \text{ }F:X\longrightarrow P(Y) \text{ }be \text{ }a \text{ }set-valued \text{ }map \text{ }and \text{ } f:X\longrightarrow Y \text{ }be \text{ }a \text{ }measurable\\ \text{ }single-valued \text{ }map, \text{ }where \text{ } F(x)=\{f(x)\}.\\ Is \text{ }F \text{ }be \text{ }a \text{ }measurable. \text{ }And \text{ }do \text{ }the \text{ }conversely \text{ }is \text{ }true .$$​

 

[mathman]

I am confused about the definition of F. It looks like it is defined for points of X and the images are points of Y considered as one point sets {f(x)} rather than the point itself f(x).

 

[moh salem]

Definition(set-valued map):
Let X and Y be two nonempty sets and P(Y)={A:A⊆Y,A≠φ}. A set-valued map is a map F:X→P(Y) i.e. ∀x∈X, F(x)⊆Y
for examples,
(1) Let F:ℝ→P(ℝ) s.t. F(x)=]α,∞[,∀x∈X. Then F is a set-valued map.
(2) Let F:ℝ→P(ℝ²) s.t. F(x)={(x,y):y=αx, α∈ℝ}.Then F is a set-valued map.
(3) Let f:ℝ→ℝ be a single-valued map defined by f(x)=x². Then F:ℝ→P(ℝ) defined by F(x)=f⁻¹(x) is a set-valued map.

 

[mathman]

To answer your original question: The definitions of measurability and integrability need to be defined for set to set functions and for point to point functions. The differences are intrinsic in that they refer to different objects.

In my experience, measure is a set function, but integrability refers to point functions.

 

[moh salem]

Thanks mathman.