# Measurability and integration of set-valued maps

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.

## Answers and Replies

mathman
Science Advisor
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.

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
Science Advisor
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).

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
Science Advisor
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.

Thanks mathman.