CHAPTER 1

The elementary theory of Q-valued functions

This chapter consists of three sections. The first one introduces a recurrent

theme: decomposing Q-valued functions in simpler pieces. We will often build on

this and prove our statements inductively on Q, relying ultimately on well-known

properties of single-valued functions. Section 1.2 contains an elementary proof of

the following fact: any Lipschitz map from a subset of

Rm

into AQ can be ex-

tended to a Lipschitz map on the whole Euclidean space. This extension theorem,

combined with suitable truncation techniques, is the basic tool of various approx-

imation results. Section 1.3 introduces a notion of differentiability for Q-valued

maps and contains some chain–rule formulas and a generalization of the classical

theorem of Rademacher. These are the main ingredients of several computations

in later sections.

1.1. Decomposition and selection for Q-valued functions

Given two elements T ∈ AQ1

(Rn)

and S ∈ AQ2

(Rn),

the sum T + S of the

two measures belongs to

AQ(Rn)

= AQ1+Q2

(Rn).

This observation leads directly

to the following definition.

Definition 1.1. Given finitely many Qi-valued functions fi, the map f1 +f2 +

. . . + fN defines a Q-valued function f, where Q = Q1 + Q2 + . . . + QN . This will

be called a decomposition of f into N simpler functions. We speak of Lebesgue

measurable (Lipschitz, H¨ older, etc.) decompositions, when the fi’s are measurable

(Lipschitz, H¨ older, etc.). In order to avoid confusions with the summation of vectors

in

Rn,

we will write, with a slight abuse of notation,

f = f1 + . . . + fN .

If Q1 = . . . = QN = 1, the decomposition is called a selection.

Proposition 0.4 ensures the existence of a measurable selection for any mea-

surable Q-valued function. The only role of this proposition is to simplify our

notation.

1.1.1. Proof of Proposition 0.4. We prove the proposition by induction on

Q. The case Q = 1 is of course trivial. For the general case, we will make use of

the following elementary observation:

(D) if

i∈N

Bi is a covering of B by measurable sets, then it suﬃces to find a

measurable selection of f|Bi∩B for every i.

Let first A0 ⊂ AQ be the closed set of points of type Q P and set B0 =

f −1(A0). Then, B0 is measurable and f|B0 has trivially a measurable selection.

Next we fix a point T ∈ AQ \ A0, T =

∑

i

Pi . We can subdivide the set of

indexes {1,...,Q} = IL ∪ IK into two nonempty sets of cardinality L and K, with

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