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