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（補足の英文資料）
https://en.wikipedia.org/wiki/Choice_function
A choice function (selector, selection) is a mathematical function f that is defined on some collection X of nonempty sets and assigns to each set S in that collection some element f(S) of S. In other words, f is a choice function for X if and only if it belongs to the direct product of X.

An example
Let X = { {1,4,7}, {9}, {2,7} }. Then the function that assigns 7 to the set {1,4,7}, 9 to {9}, and 2 to {2,7} is a choice function on X.

Choice function of a multivalued map
Given two sets X and Y, let F be a multivalued map from X and Y (equivalently, F:→ P(Y) is a function from X to the power set of Y).
A function f:→ Y is said to be a selection of F, if:
∀ x∈ X,(f(x)∈ F(x)),.
The existence of more regular choice functions, namely continuous or measurable selections is important in the theory of differential inclusions, optimal control, and mathematical economics.[2] See Selection theorem.

https://en.wikipedia.org/wiki/Selection_theorem
Selection theorem

Preliminaries
Given two sets X and Y, let F be a multivalued map from X and Y. Equivalently, F:→ P(Y) is a function from X to the power set of Y.

A function f:→ Y is said to be a selection of F if
∀ x∈ X:,,,f(x)∈ F(x),.
In other words, given an input x for which the original function F returns multiple values, the new function f returns a single value. This is a special case of a choice function.

The axiom of choice implies that a selection function always exists; however, it is often important that the selection have some "nice" properties, such as continuity or measurability. This is where the selection theorems come into action: they guarantee that, if F satisfies certain properties, then it has a selection f that is continuous or has other desirable properties.

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