Thurday, May 4, 2017
Spectral and Representation Theory
(Master Mathematik, TMP)
— Problem Set No. 2 —
Exercise 1
Remember that a function F : R → R is right continuous if limn F (xn ) = F (x) holds for every x ∈ R
and for every sequence (xn )n∈N in R with xn > x for all n ∈ N converging to x. Furthermore, F is
increasing if x ≤ y implies F (x) ≤ F (y) for all x, y ∈ R. In this excercise it may be used without proof
that for every increasing, right continous function F : R → R there is a unique Borel measure µF on R
with µ(]a, b]) = F (b) − F (a) for all a, b ∈ R with a < b, and that conversely every locally finite measure
on R is of this form.
(a) Let µ be a locally finite measure on R. Show that
)
(∞
∞
[
X
]an , bn [ , where an , bn ∈ R, an < bn
µ(E) = inf
µ(]an , bn [) E ⊆
n=1
n=1
holds for every Borel subset E of R.
(b) Show that every locally finite measure on R is inner and outer regular (and so in particular a
Radon measure).
Exercise 2
A topological group is a group (G, ·) where G carries a topological structure such that the maps · :
G × G → G and ι : G → G, g 7→ g −1 are both continuous. Here G × G is equipped with the product
topology; for every (g, h) ∈ G × G a basis of neighborhoods of (g, h) is given by the subsets of the form
M × N , where M denotes a neighborhood of g and N is a neighborhood of h. A subset S ⊆ G is
called symmetric if S = S −1 , where S −1 = {g −1 | g ∈ S}. For arbitrary subsets S, T ⊆ G, we define
ST = {gh | g ∈ S, h ∈ T }. Prove the following assertions.
(a) For every g ∈ G the left translation map τg` : G → G, h 7→ gh and the right translation map
τgr : G → G, h 7→ hg are homeomorphisms.
(b) For every open neighborhood U of the neutral element e ∈ G the inclusions Ū ⊆ U −1 U , Ū ⊆ U U −1
hold, where Ū denotes the topological closure of U .
(c) For every neighborhood N of e there is an open symmetric neighborhood V of e with V V ⊆ N .
(d) Every topological group is regular as a topological space.
Exercise 3
A Borel measure µ on a topological group G is called left-invariant if µ(gE) = µ(E) holds for every
g ∈ G and every Borel subset E of G. A Haar measure on G is a non-zero, left-invariant Radon measure.
(a) Let µ be a Haar measure on G. Show that µ(C) < +∞ holds for every compact subset C ⊆ G,
and that µ(U ) > 0 holds for every non-empty open subset U ⊆ G.
(b) Show that µ(G) < +∞ holds if and only if G is compact.
These excercises will be discussed in the tutorial on Tuesday, May 9.