Real and Complex Analysis (Rudin)

INTEGRATION ON PRODUCT SPACES 175
S Let M be the Banach space of all complex Borel measures on Rl. The norm in M is 111111 = 11l1(R 1 ).
Associate to each Borel set E c: R 1 the set
E2 = {(x, y): x + Y E E} c: R2.
If II and A E M, define their convolution II * A to be the set function given by
for every Borel set E c: Rl; II x A is as in Definition 8.7.
(a) Prove that II * A E M and that 1111 * All :5 Ii II II IIAII.
(b) Prove that II • A is the unique v E M such that
f 1 dv = If I(x + y) dJl(x) dA(y)
for every 1 E C O (Rl). (All integrals extend over Rl.)
(c) Prove that convolution in M is commutative, associative, and distributive with respect to
addition.
(d) Prove the formula
for every II and A E M and every Borel set E. Here
E - t = {x - t: x E E}.
(e) Define II to be discrete if II is concentrated on a countable set; define II to be continuous if
Jl({x}) = 0 for every point x E Rl; let m be Lebesgue measure on Rl (note that m ¢ M). Prove that
II * A is discrete if both II and A are discrete, that II • A is continuous if II is continuous and A E M,
and that II • A ~ m if II ~ m.
(f)Assume dll=ldm, dA=gdm, IEIJ(R 1 ), and gEIJ(R 1 ), and prove that
d{Jl * A) = (I. g) dm.
(g) Properties (a) and (c) show that the Banach space M is what one calls a commutative Banach
algebra. Show that (e) and (f) imply that the set of all discrete measures in M is a subalgebra of M,
that the continuous measures form an ideal in M, and that the absolutely continuous measures
(relative to m) form an ideal in M which is isomorphic (as an algebra) to IJ(Rl).
(h) Show that M has a unit, i.e., show that there exists a ~ E M such that ~ • II = II for all
liE M.
(i) Only two properties of Rl have been used in this discussion: Rl is a commutative group
(under addition), and there exists a translation invariant Borel measure m on Rl which is not identically
0 and which is finite on all compact subsets of R 1. Show that the same results hold if R 1 is
replaced by Rk or by T (the unit circle) or by Tk (the k-dimensional torus, the cartesian product of k
copies of T), as soon as the definitions are properly formulated.
6 (Polar coordinates in Rk.) Let Sk-l be the unit sphere in Rk, i.e., the set of all u E Rk whose distance
from the origin 0 is I. Show that every x E Rk, except for x = 0, has a unique representation of the
form x = ru, where r is a positive real number and u E Sk-l' Thus Rk - {O} may be regarded as the
cartesian product (0, (0) x Sk-l'
Let mk be Lebesgue measure on Rk, and define a measure uk_Ion Sk-l as follows: If A c: Sk-l
and A is a Borel set, let A be the set of all points ru, where 0 < r < I and u E A, and define