Gruber P. Convex and Discrete Geometry

398 Geometry of Numbers
Measure on E d /L
o
o
x
Fig. 26.2. Neighbourhood of a point
Fig. 26.3. Translation of a set modulo L
M
M + x
M + L x
E d /L is a compact Abelian group. Hence there is a unique (complete) Haar measure
m on it which is normalized such that m(E d /L) = d(L).
Let F be a fundamental parallelotope of L. For Lebesgue measurable sets S in F,
let V (S) denote the Lebesgue measure of S. This makes F into a measurable space
with measure V (·). It is not difficult to see that V (·) is invariant with respect to
addition modulo L in F (Fig. 26.3).
The mapping x → x∩ F of E d /L onto F is both an isomorphism and an isometry.
The Haar measures on E d /L and on F are both unique and normalized such that
E d /L and F both have measure d(L). This leads to the following result.
Proposition 26.4. AsetS ⊆ E d /L is measurable in E d /L if and only if the set
S = F ∩ S is measurable in F. In the case of measurability m(S) = V (S).
Proof. Left to the reader. ⊓⊔
26.2 The Sum Theorem of Macbeath–Kneser
Given an additive group and two of its subsets, it is a natural question to investigate
properties of their sum. For example, if the group is finite, one may ask for estimates
of the number of elements of the sum-set in terms of the numbers of elements of
the given sets. Or, if the group is endowed with a measure, one may try to relate the
measure of the sum-set to the measures of the given sets. Such problems of α+β type
were considered in additive number theory, in the theory of congruences, in group