Appendix A.
Measure
and integration
We suppose the reader is familiar with the basic facts concerning set theory and integration as they are presented in the introductory course of analysis. In this appendix, we review them briefly, and add some more which we shall need in the text. Basic references for proofs and a detailed exposition are,e.g., [[ Hal 1 ]], [[ Jar 1,2 ]], [[ KF 1,2 ]], [[ LiL ]], [[ Ru 1 ]], or any other textbook on analysis you might prefer.
A.1
Sets, mappings, relations
A set is a collection of objects called elements. The symbol cardX denotes the cardi-nality of the set X. The subset M consisting of the elements of X which satisfy the conditions P1(x), . . . , Pn(x) is usually written as M ={x∈ X: P1(x), . . . , Pn(x)}. A
set whose elements are certain sets is called asystemorfamilyof these sets; the family of all subsystems of a given X is denoted as 2X.
The operations of union, intersection, and set difference are introduced in the standard way; the first two of these are commutative, associative, and mutually distributive. In a system {Mα} of any cardinality, thede Morgan relations,
X\) α Mα = * α (X\Mα) and X\ * α Mα = ) α (X\Mα),
are valid. Another elementary property is the following: for any family {Mn}, which is at most countable, there is a disjoint family {Nn} of the same cardinality such that Nn ⊂Mn and
nNn =
nMn. The set (M \N)∪(N\M) is called thesymmetric
differenceof the setsM, N and denoted asM#N. It is commutative, M#N=N#M, and furthermore, we haveM#N= (M∪N)\(M∩N) andM#N = (X\M)#(X\N) for anyX⊃M∪N. The symmetric difference is also associative,M#(N#P) = (M#N)#P, and distributive with respect to the intersection, (M#N)∩P= (M∩P)#(N∩P).
A family R is called aset ringif M#N∈ R and M∩N∈ R holds for any pair M, N ∈ R. The relation M\N = (M#N)∩M also gives M\N ∈ R, and this in turn implies ∅ ∈ R and M∪N ∈ R. If the symmetric difference and intersection are understood as a sum and product, respectively, then a set ring is a ring in the sense of the general algebraic definition of Appendix B.1.
A.1.1 Example:Let Jd be the family of all bounded intervals in Rd, d≥1. The family Rd, which consists of all finite unions of intervals J⊂ Jd together with the empty set, is
a set ring, and moreover, it is the smallest set ring containing Jd. As mentioned above, any R∈ Rd can be expressed as a finite union ofdisjointbounded intervals.
A set ring R ⊂2X is called aset field if it contains the setX (notice that in the
terminology of Appendix B.1 it is a ring with a unit element butnotan algebra). A set field
A ⊂2X is called aσ–fieldif∞
n=1Mn∈ Aholds for any countable system {Mn}∞n=1⊂ A. De Morgan relations show that also∞n=1Mn∈ A, and furthermore that a familyA ⊂2X
containing the setX is aσ–fieldiff X\M∈ A for allM∈ A and nMn∈ A for any
at most countable subsystem {Mn} ⊂ A. Given a family S ⊂2X we consider allσ–fields A ⊂2X containing S (there is at least one,A= 2X). Their intersection is again aσ–field
containing S; we call it theσ–field generated by S and denote it as A(S).
A.1.2 Example:The elements of Bd:=A(Jd) are calledBorel setsinRd. In particular, all the open and closed sets, and thus also the compact sets, are Borel. Theσ–field Bd
is also generated by other systems,e.g., by the system of all open sets in Rd. In general,
Borel sets in a topological space (X, τ) are defined as the elements of theσ–field A(τ). A sequence {Mn}∞n=1 isnondecreasingor nonincreasingif Mn ⊂ Mn+1 or Mn ⊃
Mn+1, respectively, holds for n= 1,2, . . .. A set family M ismonotonicif it contains the set nMn together with any nondecreasing sequence {Mn}, and
nMn together
with any nonincreasing sequence{Mn}. Anyσ–field represents an example of a monotonic system. To any S there is the smallest monotonic system M(S) containing S and we haveM(S)⊂ A(S). If Ris a ring, the same is true for M(R); in addition, if M∈RM∈
M(R), then M(R) is aσ–field and M(R) =A(R).
Amapping(ormap) f from a set X toY is a rule, which associates with any x∈X a unique elementy≡f(x) of the setY; we write f:X→Y and alsox→f(x). IfY =R or Y =C the map f is usually called a real or a complexfunction, respectively. It is also often useful to consider maps which are defined on a subset Df⊂X only. The symbol
f:X→Y must then be completed by specifying the set Df which is called thedomain
off; we denote it also as D(f). If Df is not specified, it is supposed to coincide with X.
The sets Ranf :={y∈Y : y=f(x), x∈Df} and Kerf :={x∈Df: f(x) = 0} are
therangeandkernelof the map f, respectively.
A map f:X→Y isinjectiveif f(x) =f(x) holds for any x, x∈X only if x=x; it issurjectiveif Ranf =Y. A map which is simultaneously injective and surjective is calledbijectiveor abijection. The sets X and Y have the same cardinality if there is a bijection f :X→Y with Df =X. The relation f =g between f :X →Y and
g:X→Y means by definition Df=Dg and f(x) =g(x) for all x∈Df. If Df⊃Dg
and f(x) =g(x) holds for all x∈Dg we say that f is anextensionof g while g is a
restrictionof f to the set Dg; we write f⊃g and g=f|\Dg.
A.1.3 Example:For any X ⊂M we define a real function χM : χM(x) = 1 if x∈
M, χM(x) = 0 ifx∈X\M; it is called thecharacteristic(orindicator) function of the
set M. The map M →χM is a bijection of the system 2X to the set of all functions
f:X→R such that Ranf={0,1}. The setf(−1)(N) :={x∈D
f:f(x)∈N} for givenf:X→Y and N⊂Y is called
thepull–backof the set N by the map f. One has f(−1)
α∈INα
=α∈If(−1)(Nα) for
any family {Nα} ⊂Y, and the analogous relation is valid for intersections. Furthermore, f(−1)(N
1\N2) =f(−1)(N1)\f(−1)(N2) and f(f(−1)(N) = Ranf∩N. On the other hand, f(−1)(f(M))⊃M; the inclusion turns to identity iff is injective.
A.1.4 Example:Let f :X→Y with Df=X. To aσ–field B ⊂2Y we can construct
A.1 Sets, mappings, relations 597
A ⊂2X is aσ–field, then the same is true for {N⊂Y :f(−1)(N)∈ A }. Hence for any family S ⊂2Y we can construct theσ–field f(−1)(A(S))⊂2X and the latter coincides
with theσ–field generated by f(−1)(S),i.e.,f(−1)(A(S)) =A(f(−1)(S)).
Given f :X→Y and g:Y →Z we can define thecomposite map g◦f :X→Z with the domain D(g◦f) :=f(−1)(D
g) =f(−1)(Dg∩Ranf) by (g◦f)(x) :=g(f(x)). We
have (g◦f)(P)(−1)=f(−1)(g(−1)(P)) for any P ⊂Z.
If f :X→Y is injective, then for any y∈Ranf there is just one xy ∈Df such
that y =f(xy); the prescription y →g(y) :=xy defines a map g :Y →X, which is
called theinverseof f and denoted as f−1. We have D(f−1) = Ranf,Ranf−1=D
f
and f−1(f(x)) =x, f(f−1(y)) =y for any x∈D
f and y∈Ranf, respectively. These
relations further imply f(−1)(N) =f−1(N) for any N⊂Ranf. Often we have a pair of mappings f:X→Y and g:Y →X and we want to know whether f is invertible and f−1=g; this is true if one of the following conditions is valid:
(i) Dg= Ranf and g(f(x)) =x for all x∈Df
(ii) Ranf ⊂Dg, g(f(x)) =x for all x ∈Df and Rang⊂Df, f(g(y)) =y for all
y∈Dg
If f:X→Y is injective, then f−1 is also injective and (f−1)−1=f. If g:Y →Z is also injective, then the composite map g◦f is invertible and (g◦f)−1=f−1◦g−1.
TheCartesian product M×N is the set of ordered pairs [x, y] with x ∈M and y∈N; the Cartesian product of the families S and S is defined by S × S:={M×N: M ∈ S, N ∈ S}. For instance, the systems of bounded intervals of Example 1 satisfy
Jm+n
=Jm× Jn. If M×N is empty, then either M =∅ or N =∅. On the other hand, if M×N is nonempty, then the inclusion M×N ⊂P×R implies M ⊂P and N⊂R. We have (M∪P)×N= (M×N)∪(P×N) and similar simple relations for the intersection and set difference. Notice, however, that (M×N)∪(P×R) can be expressed in the form S×T only if M =P or N=R.
The definition of the Cartesian product extends easily to any finite family of sets. Alternatively, we can interpretM1×· · ·×Mnas the set of mapsf:{1, . . . , n} →
n j=1Mj
such that f(j)∈Mj. This allows us to define Xα∈IMα for a system {Mα: α∈I} of
any cardinality as the set of maps f : I →α∈IMα which fulfil f(α)∈ Mα for any
α∈I. The existence of such maps is related to the axiom of choice (see below).
Given f :X →C and g :Y → C, we define the function f×g on X×Y by (f×g)(x, y) :=f(x)g(y). Let M ⊂X×Y; then to any x∈X we define thex–cutof the set M by Mx:={y∈Y : [x, y]∈M}; we define they–cuts analogously.
Let A ⊂2X,B ⊂ 2Y beσ–fields; then theσ–field A(A × B) is called the direct
productof the fields A and B and is denoted as A ⊗ B.
A.1.5 Example:The Borel sets in Rm and Rn in this way generate all Borel sets in
Rm+n,i.e., we have Bm⊗ Bn=Bm+n.
On the other hand, the cuts of a set M∈ A ⊗ B belong to the original fields: we have Mx∈ B and My∈ A for any x∈X and y∈Y, respectively.
A subset Rϕ⊂X×X defines arelation ϕ on X: if [x, y]∈Rϕ we say the element
x is in relation with y and write x ϕ y. A common example is anequivalence, which is a relation ∼ on X that isreflexive(x∼x for any x∈X),symmetric(x∼y implies y∼x), andtransitive(x∼y and y∼z imply x∼z). For any x∈X we define the
equivalence classof x as the set Tx:={y∈X:y∼x}. We have Tx=Ty iff x∼y, so
the set X decomposes into a disjoint union of the equivalence classes.
Another important example is apartial orderingon X, which means any relation ≺ that is reflexive, transitive, andantisymmetric,i.e., such that the conditions x≺y and y≺x imply x=y. If X is partially ordered, then a subset M ⊂X is said to be(fully) orderedif any elements x, y∈M satisfy either x≺y or xy. An element x∈X is an upper boundof a set M⊂X if y≺x holds for all y∈M; it is amaximal elementof M if for any y∈M the condition yx implies y=x.
A.1.6 Theorem(Zorn’s lemma): Let any ordered subset of a partially ordered set X have an upper bound; thenX contains a maximal element.
Zorn’s lemma is equivalent to the so–calledaxiom of choice, which postulates for a system {Mα : α ∈ I} of any cardinality the existence of a map α → xα such that
xα ∈ Mα for all α ∈ I — see, e.g., [[ DS 1 ]], Sec.I.2, [[ Ku ]], Sec.I.6. Notice that the
maximal element in a partially ordered set is generally far from unique.
A.2
Measures and measurable functions
Let us have a pair (X,A), where X is a set andA ⊂2X aσ–field. A functionf:X→R
is calledmeasurable(with respect to A) if f(−1)(J)∈ A holds for any bounded interval J ⊂ R,i.e., f(−1)(J) ⊂ A. This is equivalent to any of the following statements: (i) f(−1)((c,∞))∈ A for all c∈R, (ii) f(−1)(G)∈ A for any open G⊂R, (iii) f(−1)(B)⊂
A. If X is a topological space, a function f :X→R is calledBorelif it is measurable w.r.t. theσ–field B of Borel sets in X.
A.2.1 Example:Any continuous function f :Rd →R is Borel. Furthermore, let f :
X→R be measurable (w.r.t. some A) and g:R→R be Borel; then the composite function g◦f is measurable w.r.t. A.
If functions f, g : X → R are measurable, then the same is true for their linear combinations af+bg and product f g as well as for the function x→(f(x))−1 provided f(x)= 0 for all x∈X. Even if the last condition is not valid, the function h, defined by h(x) := (f(x))−1 if f(x)= 0 and h(x) := 0 otherwise, is measurable. Furthermore, if a sequence {fn} converges pointwise, then the function x→limn→∞fn(x) is again
measurable.
The notion of measurability extends to complex functions: a function ϕ :X →C ismeasurable (w.r.t.A) if the functions Reϕ(·) and Imϕ(·) are measurable; this is trueiff ϕ(−1)(G)∈ A holds for any open set G⊂C. A complex linear combination of measurable functions is again measurable. Furthermore, if ϕ is measurable, then |ϕ(·)| is also measurable. In particular, the modulus of a measurable f:X→R is measurable, as are the functions f±:= 12(|f| ±f).
A functionϕ:X→C issimple(σ–simple) if ϕ=nynχMn, where yn∈C and the
setsMn∈ Aform a finite (respectively, at most countable) disjoint system with
nMn=
X. By definition, any such function is measurable; the sets of (σ–)simple functions are closed with respect to the pointwise defined operations of summation, multiplication, and scalar multiplication. The expression ϕ=nynχMn is not unique, however, unless the
A.2 Measures and measurable functions 599
A.2.2 Proposition:A function f :X→R is measurableiff there is a sequence {fn} ofσ–simple functions, which converges to f uniformly on X. If f is bounded, there is a sequence of simple functions with the stated property.
In fact, the approximating sequence {fn} can be chosen even to benondecreasing. If f is not bounded it can still be approximated pointwise by a sequence of simple functions, but not uniformly.
Given (X,A) and (Y,B) we can construct the pair (X×Y,A ⊗ B). Let ϕ:M→C be a function on M ∈ A ⊗ B; then itsx–cut is the function ϕx defined on Mx by
ϕx(y) :=ϕ(x, y); we define they–cut similarly. Cuts of a measurable functions may not
be measurable in general, however, it is usually important to ensure measurability a.e. – cf.Theorem A.3.13 below.
A mapping λ defined on a set family S and such that λ(M) is either non–negative or λ(M) = +∞for any M∈ S is called (a non–negative)set function. It ismonotonicif M⊂N implies λ(M)≤λ(N),additiveif λ(M∪N) =λ(M) +λ(N) for any pair of sets such that M∪N ∈ S and M∩N=∅, andσ–additiveif the last property generalizes, λ(nMn) =
nλ(Mn), to any disjoint at most countable system {Mn} ⊂ S such that
nMn∈ S.
A set function µ, which is defined on a certain A ⊂2X, isσ–additive, and satisfies
µ(∅) = 0 is called a (non–negative)measureon X. If at least oneM∈ A has µ(M)<∞, then µ(∅) = 0 is a consequence of the σ–additivity. The triplet (X,A, µ) is called a measure space;the sets and functions measurable w.r.t. A are in this case often specified asµ–measurable. A set M ∈ A is said to beµ–zeroif µ(M) = 0, a proposition–valued function defined on M ∈ A is validµ–almost everywhereif the set N ⊂M, on which it is not valid, isµ–zero. A measureµ iscompleteif N⊂M implies N∈ A for anyµ–zero set M; below we shall show that any measure can be extended in a standard way to a complete one.
Additivity implies that any measure is monotonic, and µ(M∪N) =µ(M) +µ(N)− µ(M ∩N) ≤ µ(M) +µ(N) for any sets M, N ∈ A which satisfy µ(M ∩N) < ∞. Using theσ–additivity, one can check that limk→∞µ(Mk) =µ(
∞
n=1Mn) holds for any
nondecreasingsequence {Mn} ⊂ A, and a similar relation with the union replaced by intersection is valid fornonincreasingsequences. A measure µ is said befiniteif µ(X)<
∞ andσ–finiteif X=∞n=1Mn, where Mn∈ A and µ(Mn)<∞ for n= 1,2, . . ..
Let (X, τ) be a topological space, in which any open set can be expressed as a count-able union of compact sets (as, for instance, the space Rd; recall that an open ball there
is a countable union of closed balls). Suppose thatµ is a measure on X with the domain
A ⊃τ; then the following is true: if any point of an open setGhas aµ–zero neighborhood, then µ(G) = 0.
Given a measure µwe can define the function µ:A×A →[0,∞) by µ(M×N) :=
µ(M#N). The condition µ(M#N) = 0 defines an equivalence relation on A and µ
is a metric on the corresponding set of equivalence classes.
A point x ∈ X such that the one–point set {x} belongs to A and µ({x}) = 0 is called a discrete point of µ; the set of all such points is denoted as Pµ. If µ is
σ–finite the set Pµ is at most countable. A measure µ is discrete if Pµ ∈ A and
A measureµ is said to beconcentrated on a set S∈ A if µ(M) =µ(M∩S) for any M∈ A. For instance, a discrete measure is concentrated on the set of its discrete points. If (X, τ) is a topological space and τ⊂ A, then thesupportof µ denoted as suppµ is the smallest closed set on which µ is concentrated.
Next we are going to discuss some ways in which measures can be constructed. First we shall describe a construction, which starts from a given non–negativeσ–additive set function ˙µ defined on a ring R ⊂2X; we assume that there exists an at most countable
disjoint system {Bn} ⊂ R such that nBn=X and ˙µ(Bn)<∞ for n= 1,2, . . ..
Let S be the system of all at most countable unions of the elements ofR; it is closed w.r.t. countable unions and finite intersections, andM\R∈ Sholds for allM∈ SandR∈
R. AnyM ∈ S can be expressed asM=jRj, where{Rj} ⊂ R is an at most countable
disjoint system; using it we can define the set function ¨µon S by ¨µ(M) :=jµ(R˙ j). It is
monotonic andσ–additive. Furthermore, we have ¨µ(nMn)≤
nµ(M¨ n); this property
is calledcountable semiadditivity. Together with the monotonicity, it is equivalent to the fact that M⊂∞k=1Mk implies ¨µ(M)≤
∞ k=1µ(M¨ k).
The next step is to extend the function ¨µ to the whole system 2X by defining the
outer measureby µ∗(A) := inf{µ(M¨ ) : M ∈ S, M ⊃A}. The outer measure is again monotonic and countably semiadditive; however, it is not additive so it isnota measure. Its importance lies in the fact that the system
Aµ := { A ⊂X: inf
M∈Sµ
∗(A#M) = 0}
is aσ–field. This finally allows us to define µ:= µ∗|\Aµ; it is a complete σ–additive measure on theσ–field Aµ⊃ A(R), which is determined uniquely by the set function ˙µin the sense that any measureν on A(R), which is an extension of ˙µ, satisfies ν=µ|\A(R). The measure µ is called theLebesgue extensionof ˙µ.
A measureµon a topological space (X, τ) is calledBorelif it is defined onB ≡ A(τ) and µ(C) < ∞ holds for any compact set C. We are particularly interested in Borel measures on Rd, where the last condition is equivalent to the requirement µ(K)<∞ for
any compact interval K⊂Rd.
Any Borel measure on Rd is thereforeσ–additive and corresponds to a unique σ–
additive set function ˙µ on Rd. The space Rd, however, has the special property that
for any bounded interval J∈ Jd we can find a nonincreasing sequence of open intervals
In⊃J and a nondecreasing sequence of compact intervals Kn⊂J such that
nIn=
nKn=J. This allows us to replace the requirement ofσ–additivity by the condition
˙
µ(J) = inf{µ(I) :˙ I∈ GJ}= sup{µ(K) :˙ K∈ FJ} for any J∈ Jd, where GJ ⊂ Jd is
the system of all open intervals containingJ, and FJ ⊂ Jd is the system of all compact
intervals contained inJ. A set function ˜µ on Jd which is finite, additive, and fulfils the
last condition is calledregular.
A.2.3 Theorem:There is a one–to–one correspondence between regular set functions ˜µ and µ:=µ∗|\Bd on Rd. In particular, Borel measures µ and ν coincide if µ(J) =ν(J)
holds for any J∈ Jd.
A.2.4 Example:Let f :R→R be a nondecreasing right–continuous function. For any a, b∈R, a < b, we set ˜µf(a, b) :=f(b−0)−f(a), ˜µf(a, b] :=f(b)−f(a), and analogous
A.3 Integration 601
J1; the corresponding Borel measureµ
f is called theLebesgue–Stieltjes measuregenerated
by the function f. In particular, if f is the identical function, f(x) = x, we speak about theLebesgue measureon R. Let us remark that the Lebesgue–Stieltjes measure is sometimes understood as a Lebesgue extension with a domain which is generally dependent on f; however, it contains B in any case.
A.2.5 Example:Let ˜µ and ˜ν be regular set functions on Jm ⊂Rm and Jn ⊂Rn,
respectively; then the function ˜ on Jm+n defined by ˜(J×L) := ˜µ(J)˜ν(L) is again
regular; the corresponding Borel measure is called thedirect productof the measures µ and ν which correspond to ˜µand ˜ν, respectively, and is denoted as µ⊗ν. In particular, repeating the procedure d times, we can in this way construct theLebesgue measureon Rd which associates its volume with every parallelepiped.
A.2.6 Proposition:Any Borel measure on Rd isregular,i.e., µ(B) = inf{µ(G) : G⊃
B, Gopen}= sup{µ(C) :C⊂B, Ccompact}.
As a consequence of this result, we can find to any B∈ Bd a nonincreasing sequence
of open sets Gn ⊃ B and a nondecreasing sequence of compact sets Cn ⊂B (both
dependent generally on the measure µ) such that µ(B) = limn→∞µ(Gn) =µ(
nGn) =
limn→∞µ(Cn) =µ(
nCn). Proposition A.2.6 generalizes to Borel measures on a locally
compact Hausdorff space, in which any open set is a countable union of compact sets – see [[ Ru 1 ]], Sec.2.18.
Let us finally remark that there are alternative ways to construct Borel measures. One can use,e.g., theRiesz representation theorem, according to which Borel measures correspond bijectively to positive linear functionals on the vector space of continuous functions with a compact support —cf.[[ Ru 1 ]], Sec.2.14; [[ RS 1 ]], Sec.IV.4.
A.3
Integration
Now we shall briefly review the Lebesgue integral theory on a measure space (X,A, µ). It is useful from the beginning to consider functions which may assume infinite values; this requires to define the algebraic operations a+∞:=∞, a· ∞:=∞ for a >0 and a· ∞:= 0 for a= 0,etc., to add the requirement f(−1)(∞) ∈ A to the definition of measurability, and several other simple modifications.
Given a simple non–negative function s:= nynχMn on X, we define its integral
by Xs dµ:=nynµ(Mn); correctness of the definition follows from the additivity of µ.
In the next step, we extend it to all measurable functions f:X→[0,∞] putting
X f dµ ≡ X f(x)dµ(x) := sup X s dµ :s∈Sf ,
where Sf is the set of all simple functions s:X →[0,∞) such that s≤f. We also
define Mf dµ:=
Xf χMdµ for any M∈ A; in this way we associate with the function
f and the set M a number from [0,∞], which is called the (Lebesgue)integralof f over M w.r.t. the measure µ.
A.3.1 Proposition:Let f, g be measurable functions X →[0,∞] and M ∈ A; then
M(kf)dµ =k
Mf dµ holds for any k∈[0,∞), and moreover, the inequality f ≤g
Notice that the integral of f= 0 is zero even if µ(X) =∞. On the other hand, the relation X|ϕ|dµ= 0 for any measurable function ϕ:X→C implies that µ({x∈X: ϕ(x)= 0}) = 0,i.e., that the function ϕ is zeroµ–a.e. Let us turn to limits which play the central role in the theory of integration.
A.3.2 Theorem(monotone convergence): Let {fn} be a nondecreasing sequence of non-negative measurable functions; then limn→∞Xfndµ=
X(limn→∞fn)dµ.
The right side of the last relation makes sense since the limit function is measurable. However, we often need some conditions under which both sides are finite. The correspond-ing modification is also called themonotone–convergence(orLevi’s)theorem: if{fn}is a nondecreasing sequence of non–negative measurable functions and there is a k >0 such that Xfndµ ≤ k for n = 1,2, . . ., then the function x → f(x) := limn→∞fn(x) is
µ–a.e. finite and limn→∞Xfndµ=
Xf dµ ≤k. The monotone–convergence theorem
implies, in particular, that the integral of a measurable function can be approximated by a nondecreasing sequence of integrals of simple functions.
A.3.3 Corollary (Fatou’s lemma): X(liminfn→∞fn)dµ ≤ liminfn→∞Xfndµ holds
for any sequence of measurable functions fn:X→[0,∞].
This result has the following easy consequence: let a sequence {fn} of non–negative measurable functions have a limit everywhere, limn→∞fn(x) =f(x), and
Xfndµ≤k
for n= 1,2, . . .; then Xf dµ≤k. Applying the monotone–convergence theorem to a
se-quence{fn}of non–negative measurable functions we get the relationX(∞n=1fn)dµ= ∞
n=1
Xfndµ. In particular, if f :X →[0,∞] is measurable and {Mn}∞n=1 ⊂ A is a disjoint family with ∞n=1Mn = M, then putting fn := f χMn we get
Mf dµ = ∞
n=1
Mnf dµ. This relation is called σ–additivity of the integral; it expresses the fact
that the function f together with the measure µ generates another measure.
A.3.4 Proposition:Let f:X→[0,∞] be a measurable function; then the map M→ ν(M) :=Mf dµ is a measure with the domain A, and Xg dν=Xgf dµ holds for any measurable g:X→[0,∞].
Let us pass to integration of complex functions. A measurable function ϕ:X→C isintegrable (over X w.r.t. µ) if X|ϕ|dµ <∞ (recall that if ϕ is measurable so is
|ϕ|). The set of all integrable functions is denoted as L(X, dµ); in the same way we define
L(M, dµ) for any M∈ A. Givenϕ∈ L(X, dµ) we denote f:= Reϕ and g:= Imϕ; then f± and g± are non–negative measurable functions belonging to L(X, dµ). This allows us to define theintegralof complex functions through the positive and negative parts of the functions f, g as the mapping
ϕ −→ X ϕ dµ := X f+dµ− X f−dµ+i X g+dµ−i X g−dµ
of L(X, dµ) to C. If, in particular, µ is the Lebesgue measure on Rd we often use the
symbolL(Rd) orL(Rd, dx) instead ofL(Rd, dµ), and the integral is written asϕ(x)dx, or occasionally as ϕ(x)dx.
The above definition has the following easy consequence: if Mϕ dµ= 0 holds for
all M ∈ A, then ϕ(x) = 0 µ–a.e. in X. Similarly Mϕ dµ≥0 for all M ∈ A implies ϕ(x) ≥0 µ–a.e. in X; further generalizations can be found in [[ Ru 1 ]], Sec.1.40. The integral has the following basic properties:
A.3 Integration 603
(a) linearity: L(X, dµ) is a complex vector space and X(αϕ+ψ)dµ = αXϕ dµ+
Xψ dµ for all ϕ, ψ∈ L(X, dµ) and α∈C,
(b) Xϕ dµ≤X|ϕ|dµ holds for any ϕ∈ L(X, dµ).
A.3.5 Examples: A simple complex function σ = nηnχMn on X is integrableiff
n|ηn|µ(Mn)<∞, and in this case
Xσ dµ=
nηnµ(Mn). The same is true forσ–
simple functions. Further, let functions f :X→[0,∞] and ϕ:X→C be measurable and dν:=f dµ; then ϕ∈ L(X, dν) iff ϕf∈ L(X, dµ) and Proposition A.3.4 holds again with g replaced by ϕ.
For a finite measure, we have an equivalent definition based on approximation of integrable functions by sequences ofσ–simple functions.
A.3.6 Proposition: If µ(X) < ∞, then a measurable function ϕ : X → C belongs to L(X, dµ) iff there is a sequence {τn} of σ–simple integrable functions such that limn→∞(supx∈X|ϕ(x)−τn(x)|) = 0; if this is the case, then
Xϕ dµ= limn→∞
Xτndµ.
Moreover, if ϕ is bounded the assertion is valid with simple functions τn.
One of the most useful tools in the theory of integral is the following theorem.
A.3.7 Theorem (dominated convergence, or Lebesgue): Let M ∈ A and {ϕn} be a sequence of complex measurable functions with the following properties: ϕ(x) := limn→∞ϕn(x) exists forµ–almost all x ∈ M and there is a function ψ ∈ L(X, dµ)
such that |ϕn(x)| ≤ψ(x) holdsµ–a.e. in M for n= 1,2, . . .. Then ϕ∈ L(X, dµ) and
lim n→∞ M |ϕ−ϕn|dµ = 0, lim n→∞ M ϕndµ = M ϕ dµ.
Suppose that we have non–negative measures µ and ν on X (without loss of gener-ality, we may assume that they have the same domain) and k >0; then we can define the non–negative measure λ:=kµ+ν. We obviously have L(X, dλ) =L(X, dµ)∩ L(X, dν) and Xψ dλ=kXψ dµ+Xψ dν for any ψ∈ L(X, dλ). In particular, if non–negative measures µand λ with the same domainA satisfy µ(M)≤λ(M) for any M∈ A, then
L(X, dλ)⊂ L(X, dµ) and Xf dµ≤Xf dλ holds for each non–negative f∈ L(X, dλ). Let µ, ν again be measures on X with the same domain A. We say that ν is absolutely continuousw.r.t. µ and write ν µ if µ(M) = 0 implies ν(M) = 0 for any M ∈ A. On the other hand, if there are disjoint sets Sµ, Sν ∈ A such that µ is
concentrated on Sµ and ν on Sν we say that the measures aremutually singularand
write µ⊥ν.
A.3.8 Theorem:Let λ and µ be non–negative measures on A, the former being finite and the latterσ–finite; then there is a unique decomposition λ=λac+λs into the sum
of non–negative mutually singular measures such that λac µ and λs⊥µ. Moreover,
there is a non–negative function f ∈ L(X, dµ), unique up to aµ–zero measure subset of X, such that dλac=f dµ,i.e., λac(M) =
Mf dµ for any M∈ A.
The relationλ=λac+λs is called theLebesgue decompositionof the measure λ. The
second assertion implies theRadon–Nikod´ym theorem:letµ beσ–finite and λ finite; then λµ holdsiff there is f∈ L(X, dµ) such that dλ=f dµ.
A.3.9 Remark:There is a close connection between these results (and their extensions to complex measures mentioned in the next section) and the theory of the indefinite Lebesgue integral, properties of absolutely continuous functions,etc.We refer to the literature men-tioned at the beginning; in this book, in fact, we need only the following facts: a function ϕ:R→C isabsolutely continuouson a compact interval [a, b] if for any ε >0 there is δ >0 such that j|ϕ(βj)−ϕ(αj)|< ε holds for a finite disjoint system of intervals
(αj, βj)⊂[a, b] fulfilling
j(βj−αj)< δ. The function ϕ is absolutely continuous in
a (noncompact) interval J if it is absolutely continuous in any compact [a, b]⊂J. A function ϕ: R→ C is absolutely continuous in R iff its derivative ϕ exists almost everywhere w.r.t. the Lebesgue measure and belongs toL(J, dx) for any bounded interval J⊂R with the endpoints a≤b; in such a case we have ϕ(b)−ϕ(a) =abϕ(x)dx.
Next we shall mention integration of composite functions. Let w :X →Rd be a map such that w(−1)(Bd) ⊂ A; this requirement is equivalent to measurability of the
“component” functions wj :X→R,1≤j≤d. Suppose that µ(w(−1)(J))<∞ holds
for any J∈ Jd; then the relation B→µ(w)(B) :=µ(w(−1)(B)) defines a Borel measure µ(w) on Rd.
A.3.10 Theorem:Adopt the above assumptions, and let a Borel function ϕ:Rd→C
belong to L(Rd, dµ(w)); then ϕ◦w∈ L(X, dµ) and
B
ϕ dµ(w) =
w(−1)(B)(ϕ◦w)dµ
holds for all b∈ Bd.
In particular, ifX=Rd,A=Bd, and µ(w) is the Lebesgue measure on Rd, the latter
formula can under additional assumptions be brought into a convenient form. Suppose that w:Rd→Rd is injective, its domain is an open set D ⊂Rd, the component functions wj:D →R have continuous partial derivatives, (∂kwj)(·) for j, k= 1, . . . , d, and finally,
the Jacobian determinant, Dw:= det(∂kwj), is nonzero a.e. in D. Such a map is called
regular;its rangeR:= Ranw is an open set in Rd, and the inverse w−1 is again regular.
A.3.11 Theorem(change of variables): Let w be a regular map on Rd with the domain D and range R; then a Borel function ϕ:R →C belongs to L(R, dx) iff (ϕ◦w)Dw∈ L(D, dx); in that case we have
B
ϕ(x)dx =
w(−1)(B)
((ϕ◦w)|Dw|)(x)dx for any Borel B⊂ R.
In Example A.2.5 we have mentioned how the measure µ⊗ν can be associated with a pair of Borel µ, ν on Rd. An analogous result is valid under much more general circumstances.
A.3.12 Theorem:Let (X,A, µ) and (Y,B, ν) be measure spaces withσ–finite measures; then there is just one measure λ on X×Y with the domain A ⊗ B such thatλ(A×B) =µ(A)ν(B) holds for all A∈ A, B∈ B; this measure isσ–finite and satisfies λ(M) =
Xν(Mx)dµ(x) =
Yµ(M
A.4 Complex measures 605
The measure λ is again called theproduct measureof µ and ν and is denoted as µ⊗ν. Using it we can formulate the following important result.
A.3.13 Theorem(Fubini): Suppose the assumptions of the previous theorem are valid and ϕ :X×Y →C belongs to L(X×Y, d(µ⊗ν)). Then the cut ϕx ∈ L(Y, dν) for
µ–a.a. x ∈ M and the function Φ : Φ(x) =Yϕxdν belongs to L(X, dµ); similarly
ϕy∈ L(X, dµ) ν–a.e. in N and Ψ : Ψ(y) =Xϕydµ belongs to L(Y, dν). Finally, we have XΦdµ=YΨdν=X×Yϕ d(µ⊗ν), or in a more explicit form,
X Y ϕ(x, y)dν(y) dµ(x) = Y X ϕ(x, y)dµ(x) dν(y) = X×Y ϕ(x, y)d(µ⊗ν)(x, y).
We should keep in mind that the latter identity may not be valid if at least one of the measures µand ν is notσ–finite. It is also not sufficient that both double integrals exists — counterexamples can be found,e.g., in [[ Ru 1 ]], Sec.7.9 or [[ KF ]], Sec.V.6.3. However, if at least one double integral of themodulus |ϕ| is finite, then all the conclusions of the theorem are valid.
A.4
Complex measures
Aσ–additive map ν:A →C corresponding to a given (X,A) is calledcomplex measure on X; if ν(M)∈R for all m∈ A we speak about areal(orsigned) measure. Any pair of non–negative measuresµ1, µ2 with a common domain A determines a signed measure by := µ1−µ2; similarly a pair of real measures 1, 2 defines a complex measure by ν:=1+i2.
Any at most countable system {Mj}, which is disjoint and satisfies M = jMj,
will be called decompositionof the set M; the family of all decompositions of M will be denoted by SM. To a given complex measure ν and M ∈ A, we define |ν|(M) := supj|ν(Mj)|:{Mj} ∈ SM
. One has |ν|(M) ≥ |ν(M)|; the set function |ν|(·) is called the (total)variationof the measure ν.
A.4.1 Proposition:The variation of a complex measure is a non–negative measure; it is the smallest non–negative measure such that µ(M)≥ |ν(M)| holds for all M∈ A.
Using the total variation, we can decompose in particular any signed measure in the form =µ+−µ−, where µ±: µ±(M) = 1
2[||(M)±(M)]. Since in general the decomposition of a signed measure into a difference of non–negative measures is not unique, one is interested in the minimal decomposition =µ+
−µ− such that any pair of non–
negative measuresµ1, µ2 on A with the property =µ1−µ2 satisfies µ1(M)≥µ+(M)
and µ2(M)≥µ−(M) for each M∈ A.
The minimality is ensured if there is a disjoint decompositionQ+∪Q−=X such that µ± := ±(M∩Q±)≥0; the pair {Q+, Q−} is calledHahn decompositionof X w.r.t.
the measure . The Hahn decomposition always exists but it is not unique. Nevertheless, if {Q˜+,Q˜−} is another Hahn decomposition, one has (M∩Q±) =(M∩Q˜±) for any M ∈ A, so the measures µ± depend on only; we call them thepositiveandnegative
variation of the measure . The formula =µ+−µ− is named theJordan decomposition
One has µ+
(M) +µ−(M) =||(M) and µ±(M) = sup{ ±(A) :A⊂M, A∈ A }for
anyM∈ A. As a consequence, the positive and negative variations of a signed measure as well as the total variation of a complex measure are finite. One can introduce also infinite signed measures; however, we shall not need them in this book.
Complex Borel measureson Rd have A=Bd for the domain. Variation of a complex
Borel measure is a non–negative Borel measure. As in the non–negative case, a complex Borel measure can be approximated using monotonic sequences of compact sets from inside and open sets from outside of a given M ∈ Bd. Also the second part of Theorem A.2.3
can be generalized.
A.4.2 Proposition:Let complex Borel measure ν and ˜ν on Rd satisfy ν(J) = ˜ν(J)
for all J∈ Jd; then ν= ˜ν.
Before proceeding further, let us mention how the notion of absolute continuity extends to complex measures. The definition is the same: a complex measure ν isabsolutely continuousw.r.t. a non–negative µif µ(M) = 0 implies ν(M) = 0 for all M∈ A. There is an alternative definition.
A.4.3 Proposition:A complex measure ν satisfies νµ iff for any ε >0 there is a δ >0 such that µ(M)< δ implies |ν(M)|< ε.
In particular, ifϕ∈ L(X, dµ) and ν is the measure generated by this function, ν(M) :=
Mϕ dµ, then ν µ, so for any ε >0 there is a δ >0 such that µ(M)< δ implies
Mϕ dµ< ε; this property is calledabsolute continuity of the integral.
Theorem A.3.8 holds for a complex measure λ as well. The measure can be evenσ– finite; however, then the function f belongs no longer to L(X, dµ), it is only measurable and integrable over any set M ∈ A with λ(M) < ∞. The Radon–Nikod´ym theorem yields thepolar decomposition of a complex measure:
A.4.4 Proposition:For any complex measure ν there is a measurable function h such that |h(x)|= 1 for all x∈X and dν=h d|ν|.
Let us pass now to integration with respect to complex measures. We start with a signed measure =µ+ −µ−: a function ϕ:X→C isintegrablew.r.t. if it belongs to L(X, dµ+
)∩ L(X, dµ−) =:L(X, d). Its integral is then defined by X ϕ d:= X ϕ dµ+ − X ϕ dµ− ;
the correctness follows from the uniqueness of the Jordan decomposition. The integral w.r.t. a complex measure ν represents then a natural extension of the present definition: for any functionϕ:X→C belonging to L(X, dν) :=L(X, dReν)∩ L(X, dImν) we set
X ϕ dν := X ϕ dReν+ X ϕ dImν.
The set of integrable functions can be expressed alternatively as L(X, dν) =L(X, d|ν|); it is a complex vector space and the map ϕ→Xϕ dν is again linear. Also other properties of the integral discussed in the previous section extend to the complex–measure case. For instance, the inequality Xϕ dν ≤ X|ϕ|d|ν| holds for any ϕ ∈ L(X, dµ). We shall not continue the list, restricting ourselves by quoting the appropriate generalization of Proposition A.3.4.
A.5 The Bochner integral 607
A.4.5 Proposition:Let ϕ∈ L(X, dν) for a complex measure ν; then the map M → γ(M) :=Mϕ dν defines a complex measure γ on A. The conditions ψ∈ L(X, dγ) and ψϕ∈ L(X, dν) are equivalent for any measurable γ:X→C; if they are satisfied one has
Xψ dγ=
Xψϕ dν.
A.5
The Bochner integral
The theory of integration recalled above can be extended to vector–valued functions F: Z → X, where X is a Banach space; they form a vector space denoted as V(Z,X) when equipped with pointwise defined algebraic operations. Let (Z,A, µ) be a measure space with a positive measure µ. A function S∈ V(Z,X) issimpleif there is a disjoint decomposition {Mj}n
j=1⊂ A of the set Z and vectors y1, . . . , yn ∈ X such that S = n
j=1yjχMj. The integral of such a function is defined by
ZS(t)dµ(t) := n
j=1yjµ(Mj);
as above, it does not depend on the used representation of the function S.
To any F∈ V(Z, dµ) we define the non–negative function F:=F(·). A vector– valued function F is integrable w.r.t. µ if there is a sequence {Sn} of simple functions such that Sn(t)→F(t) holds for µ–a.a. t∈Z and
ZF−Sndµ→0. The set of all
integrable functions F :Z→ X is denoted by B(Z, dµ;X). If F is integrable, the limit
Z
F(t)dµ(t) := lim
n→∞ Z
Sn(t)dµ(t)
exists and it is independent of the choice of the approximating sequence; we call it the Bochner integralof the function F. The function χMF is integrable for any set M∈ A
and F∈ B(Z, dµ;X), so we can also defineMF(t)dµ(t) :=
ZχM(t)F(t)dµ(t). If {Mk}
is a finite disjoint decomposition of M, we have
M
F(t)dµ(t) =
k Mk
F(t)dµ(t),
which means that the Bochner integral isadditive.
A.5.1 Proposition: The map F → ZF(t)dµ(t) from the subspace B(Z, dµ;X) ⊂
V(Z,X) to X is linear. Suppose that for a vector–valued function F there is a sequence
{Sn} of simple functions that converges to F µ–a.e.; thenF belongs to B(Z, dµ;X) iff
F ∈ L(Z, dµ), and in that case
ZF(t)dµ(t) ≤
Z
F(t)dµ(t).
The existence of an approximating sequence of simple functions has to be checked for each particular case; it is easy in some situations,e.g., if Z is a compact subinterval in R and F is continuous, or if Z is any interval, F is continuous and its one–sided limits at the endpoints exist. The continuity of F :R → X in an interval [a, b] also implies the relation dtd atF(u)du=F(t) for any t∈(a, b). Proposition A.5.1 shows that the Bochner integral isabsolutely continuous: for any ε >0 there is a δ >0 such that
NF(t)dµ(t)< ε holds for any N ∈ A with µ(N)< δ. Another useful result is the following.
A.5.2 Proposition:If B:X → Y is a bounded linear map to a Banach space Y; then the condition F ∈ B(Z, dµ;X) implies BF ∈ B(Z, dµ;Y), and
Z (BF)(t)dµ(t) = B Z F(t)dµ(t) .
Many properties of the Lebesgue integral can be extended to the Bochner integral. Probably the most important among them is the dominated–convergence theorem.
A.5.3 Theorem:Let {Fn} ⊂ B(Z, dµ;X) be a sequence such that {Fn(t)} converges
for µ–a.a. t∈Z and Fn(t) ≤g(t), n= 1,2, . . ., for someg∈ L(Z, dµ). Assume further
that there is a sequence{Sn} of simple functions, which converges to the limiting function F:F(t) = limn→∞Fn(t) µ–a.e. in Z; then F∈ B(Z, dµ;X) and
lim
n→∞ Z
Fn(t)dµ(t) = Z
F(t)dµ(t).
An analogue to Theorem A.3.10 can be proven for some classes of functions,e.g., for a monotonic w:R→R.
Since B(X) is a Banach space, the Bochner integral is also used for operator–valued functions. For instance, suppose that a mapB:R→ B(X) is such that the vector–valued function t→B(t)x is continuous for any x ∈ X. Further, let K ⊂R be a compact interval and µ a Borel measure on R; then limn→∞KB(t)xndµ(t) =
KB(t)x dµ(t)
holds for any sequence {xn} ⊂ X converging to a point x. Moreover, if an operator T∈ C commutes with B(t) for all t∈K, then KB(t)y dµ(t) belongs to D(T) for any
Appendix B.
Some algebraic
notions
In this appendix we collect some algebraic definitions and results needed in the text. There are again many textbooks and monographs in which this material is set out extensively; let us name,e.g., [[ BR 1 ]], [[ Nai 1 ]], [[ Ru 2 ]], or [[ Ti ]] for associative algebras, and [[ BaR ]], [[ Kir ]], [[ Pon ]], or [[ ˇZel ]] for Lie groups and algebras.
B.1
Involutive algebras
Abinary operationin a set M is a map ϕ:M×M→M; it isassociativeorcommutative if ϕ(ϕ(a, b), c) =ϕ(a, ϕ(b, c)) or ϕ(a, b) =ϕ(b, a) , respectively, holds for all a, b, c∈M. A set G equipped with an associative binary operation is called agroupif there exist the unit element e∈ G, ϕ(g, e) = ϕ(e, g) = g for any g ∈ G, and the inverse element
g−1∈G to any g∈G, ϕ(g, g−1) =ϕ(g−1, g) =e.
Consider next a setRequipped with two binary operations, which we callsummation, ϕa(a, b) :=a+b, andmultiplication, ϕm(a, b) :=ab. The triplet (R, ϕa, ϕm) is aringif
(R, ϕa) is a commutative group and the two operations are distributive, a(b+c) =ab+ac
and (a+b)c =ac+bc for all a, b, c∈R. If there is an e∈R such that ae=ea=a holds for all a∈R, we call it theunit elementof R.
Let A be a vector space over a field F. The vector summation gives it the structure of a commutative group; if we define a multiplication which is distributive with the sum-mation and satisfies α(ab) = (αa)b=a(αb) for any a, b∈ A, α∈C, thenA becomes a ring, which we call alinear algebraover the field F, in particular, arealorcomplexalgebra if F =R or F=C, respectively. An algebra is said to beassociativeif its multiplication is associative. The term “algebra” without a further specification always means a com-plex associative algebra in what follows; we should stress, however, that many important algebras are nonassociative,e.g., the Lie algebras discussed in Sec.B.3 below. An algebra isAbelianorcommutativeif its multiplication is commutative.
Asubalgebraof an algebra A is a subset B, which is itself an algebra with respect to the same operations. If A has the unit element, which is not contained in B, then we can extend the subalgebra to ˜B := {αe+b : α ∈ C, b ∈ B}; in a similar way, any algebra can be completed with the unit element by extending it to the set of pairs [α, a], α ∈ C, a ∈ A, with the appropriately defined operations. A proper subalgebra
B ⊂ A is called a (two–sided) idealin A if the products ab and ba belong to B for all a∈ A, b∈ B; we define theleftandrightideal analogously. A trivial example of an ideal is the zero subalgebra {0} ⊂ A. The algebra A itself is not regarded as an ideal;
thus no ideal can contain the unit element. Amaximalideal in A is such that it is not a proper subalgebra of another ideal inA; any ideal in an algebra with the unit element is a subalgebra of some maximal ideal. An algebra is calledsimpleif it contains no nontrivial two–sided ideal. The intersection of any family of subalgebras (ideals, one–sided ideals) in
A is respectively a subalgebra (ideal, one–sided ideal), while the analogous assertion for the unions isnotvalid.
LetA be an algebra with the unit element. We say that an elementa∈ Aisinvertible if there exists aninverse element a−1∈ A such that a−1a=aa−1=e; we define theleft andrightinverse in the same way. For any a∈ Athere is at most one inverse; an element is invertibleiffit belongs to no one–sided ideal of the algebra A, which means, in particular, that in an algebra without one–sided ideals any nonzero element is invertible. Recall that afieldis a ring with the unit element which has the last named property; the examples are R,C or the noncommutative field Q of quaternions.
We define thespectrumof a∈ A as the set σA(a) :={λ: (a−λe)−1does not exist}. The complement ρA(a) :=C\σA(a) is called theresolvent set;its elements areregular valuesfor which the theresolvent ra(λ) := (a−λe)−1 exists.
B.1.1 Proposition:Let A be an algebra with the unit element; then
(a) If a, ab are invertible,b is also invertible. If ab , ba are invertible, so are a and b. (b) Ifab=e, the element ba is idempotent but it need not be equal to the unit element
unless dimA<∞.
(c) If e−ab is invertible, the same is true for e−ba.
(d) σA(ab)\ {0}=σA(ba)\ {0}, and moreover, σA(ab) =σA(ba) provided one of the elements a, b is invertible.
(e) σA(a−1) ={λ−1: λ∈σ
A(a)}holds for any invertible a∈ A.
For any set S ⊂ A we define the algebra A0(S) generated by S as the smallest subalgebra in A containing S; it is easy to see that it consists just of all polynomials composed of the elements of S without an absolute term. We say thatS is commutative if ab=ba holds for any a, b∈ S; the algebra A0(S) is then Abelian. Amaximal Abelian algebra is such that it is not a proper subalgebra of an Abelian subalgebra; any Abelian subalgebra in A can be extended to a maximal Abelian subalgebra. We also define the commutantof a set S ⊂ Aas S:={a∈ A: ab=ba, b∈ S }; in particular, thecenteris the set Z:=A. We define thebicommutant S:= (S) and higher–order commutants in the same way.
B.1.2 Proposition:Let S,T be subsets in an algebra S; then
(a) S and S are subalgebras containing the centerZ, and also the unit element if A has one. Moreover, S=S=· · · and S=SIV
=· · ·. (b) The inclusion S ⊂ T implies S⊃ T.
(c) S ⊂ S, and S is commutativeiff S ⊂ S, which is further equivalent to the condition that S is Abelian.
(d) A0(S)=S and A0(S)=S.
(e) A subalgebra B ⊂ A is maximal Abelianiff B=B; in that case also B=B. Let us turn to algebras with an additional unary operation. Recall that an involution a→a∗ on a vector spaceA is an antilinear mapA → A such that (a∗)∗=aholds for all a∈ A; aninvolution on an algebrais also required to satisfy the condition (ab)∗=b∗a∗ for any a, b∈ A. An algebra equipped with an involution is called aninvolutive algebra
B.1 Involutive algebras 611
or briefly a ∗–algebra. A subalgebra in A, which is itself a ∗–algebra w.r.t. the same involution, is called a ∗–subalgebra;we define the ∗–idealin the same way. The element a∗ is said to beadjointto a. Given a subset S ⊂ A we denote S∗:= {a∗ : a∈ S}; the set S issymmetricif S∗=S; in particular, an element a fulfilling a∗=a is called Hermitean. By A∗0(S) we denote the smallest ∗–subalgebra in A containing the set S.
B.1.3 Proposition:Let A be a ∗–algebra; then
(a) Any element is a linear combination of two Hermitean elements, and e∗=e provided
A has the unit element.
(b) a∗ is invertibleiff a is invertible, and (a∗)−1= (a−1)∗. (c) σA(a∗) =σA(a) holds for any a∈ A.
(d) A subalgebra B ⊂ A is a ∗–subalgebraiff it is symmetric; the intersection of any family of ∗–subalgebras (∗–ideals) is a ∗–subalgebra (∗–ideal).
(e) Any ∗–ideal in A is two–sided.
(f) A∗0(S) =A0(S ∪ S∗) holds for any subset S ⊂ A; if S is symmetric, then S and
S are ∗–subalgebras in A.
B.1.4 Example(bounded–operator algebras): The set B(H) with the natural algebraic operations and the involution B→B∗ is a ∗–algebra whose unit element is the operator I. Let us mention a few of its subalgebras:
(a) If E is a nontrivial projection, then{EB : B∈ B(H)} is a right ideal but not a
∗–subalgebra; on the other hand, {EBE : B∈ B(H)} is a ∗–subalgebra but not an ideal.
(b) If dimH=∞, the sets K(H)⊃ J2(H)⊃ J1(H) of compact, Hilbert–Schmidt, and trace–class operators, respectively, are ideals in B(H); similarly Jp(H) is an ideal in anyJq(H), q > p,etc.
(c) The algebra A0(B) generated by an operator B∈ B(H) consists of all polynomials in B without an absolute term. It is a ∗–algebra if B is Hermitean, while the opposite implication is not valid; for instance, the Fourier–Plancherel operator F is non–Hermitean but A0(F) is a ∗–algebra because F3=F−1=F∗.
The algebras of bounded operators, which represent our main topic of interest, inspire some definitions. We have already introduced the notions of spectrum and hermiticity; similarly an elementa∈ A is said to benormalif aa∗=a∗a, aprojectionif a∗=a=a2, andunitaryif a∗=a−1,etc.Of course, we also employ other algebras than B(H) and its subalgebras,e.g., the Abelian ∗–algebra C(M) of continuous complex functions on a compact space M with natural summation and multiplication, and the involution given by complex conjugation, (f∗)(x) :=f(x) .
An idealJ in an algebraAis a subspace, so we can construct the factor space A/J. It becomes an algebra if we define on it a multiplication by ˜a˜b:=ab0, where a, b are any elements representing the equivalence classes ˜a and ˜b; it is called thefactor algebra(of
A w.r.t. the idealJ). If A has the unit element, then the class ˜e:={e+c : c∈ J } is the unit element of A/J.
Amorphismof algebras A,B is a map ϕ:A → B which preserves the algebraic structure, ϕ(αa+b) = αϕ(a) +ϕ(b) and ϕ(ab) =ϕ(a)ϕ(b) for all a, b∈ A, α ∈ C. In particular, if ϕ is surjective, then the image of the unit element (an ideal, maximal ideal, maximal Abelian subalgebra) in A is respectively the unit element (an ideal, . . . ) in B. If ϕ is bijective, we call it anisomorphism; in the case A=B one uses the terms
