Integration generated by a volume which is an infinite sum of volumes

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PEACE MATEMATY CZNE X V II (1974)

W. M. Bogdanowicz and John N. Welch (Washington, D. C.)

Integration generated by a volume which is an infinite sum of volumes

Introduction. A non-empty family of sets F of a space X is called a prering if for every two sets A ,B c V the sets A n B ,A \ B can be represented as a finite nnion of pair-wise disjoint sets from the family F.

A function v is called a volume if it is non-negative, real-valued and countably additive on a prering F. The triple (X, F, v) is called a volume space if v is a volume on the prering F.

If (У, I I) is a Banach space, denote by 8 {V, 7 ) the family of all functions of the form s = сА1у х-\- ... + елпУт where A{ is a finite family of pair-wise disjoint sets from V, the points y x, ..., yn are from the Banach space 7 , and where cA denotes the characteristic function of the set A . Members of 8 ( V, 7 ) are called simple functions.

Let В denote the set of all real numbers. Denote by $(F ) the family of all sets A a X such that cA e $(F , R). It follows immediately that the family $(F ) consists of all sets of the form A = A xu . .. u i №, where the sets А{е V (i = 1, ..., n) are pair-wise disjoint. It is also easy to see that $(F ) is a ring of sets.

The theory of the integral and of the spaces Lp {v, 7 ) (1 < oo) generated by the volume space (X , F, v) is developed in [1] and [2].

For a development of the theory of the space M (v, 7 ) of Lebesgue-Bochner measurable functions and the family M (v) of measurable sets generated by the volume v see [3].

If v is any volume on the prering F we denote by N(v) the family of all subsets А с X satisfying the condition that for each e > 0 there exists a sequence An of sets in F such that A c [ J An and ] 8 v(An) < e.

n n

Members of Ж (v) are called null-sets. A condition G (x) depending on a para­

meter » e l is said to be satisfied v-almost everywhere (t)-a. e.) if there exists a set C eX (v) such that the condition G(æ) is satisfied at every point x 4 G.

322 W. M. B ogdanow icz and J. N. W elch

By L œ (v, Г) we denote the subfamily of all functions fe Ш(v, Y) satisfying the condition that \f(x)\ < M v-a. e. for some constant M.

For f e L ^ iv , Y) we set

ll/lloo — in f{Ж: \f{x)\^M tf-a.e.}.

If Vi {i — 1, ... , n) aie volumes on a prering V, then the set function v — vx + .. .+ vn is a volume on V. In [8] we established the relations that exist between the spaces Lp(v, Y), M (v, Y), M (v) and Lp(v{, Y), M{vif Y ), M(Vj) (i = 1, ..., n) respectively.

If vt (t e T) are volumes on a prering, where T is an infinite parameter set such that f>]vt{A) < oo for all A e V, then the set function v, defined

T

by v{A) = ^r>t{A) for all sets A e V, is a volume on V, and all the relations

T

established in [8] for finite sums of volumes are valid for this case.

In Section 2 we point out the connections between the approach to integration based on volumes and that based on measures. Using these connections we formulate the results of Section 1 for measures.

1. Integration with respect to an infinite sum of volumes.

Lem m a 1. Let vt(te T) be a fam ily of volumes on V such that ]? v t(A)

T

< oo for all A t V', then the set function v, defined by = ] ? v t(A) for

all A e V, is a volume on V. T

P ro o f. Let us notice that if and S 2 are any abstract sets and if a is a mapping from 8 г x $ 2 into the extended real numbers, then we have the equality

sup a (s 1} s2) = sup sup a (sa, s2) = sup sup a («j, s2).

S i X S2 S i S2 S2 S i

Prom the previous we get

£ v ( A n) = e ( E v‘ <~M ^ E i E ^ M -

n n T T n

Since each vt is a volume on V we have ]£vt (An) = vt {A) yielding I > ( A ) = 2 M A ) = v(A )-

n T

Theorem 1. Let vt (teT ) be volumes on the prering V such that JT vt(A)

T

< oo for all A eV . Then for the volume v = on the prering V we have :

T

(a) The fam ily N (v) of v-null sets is the intersection of the fam ilies

* ( v t).

(b) For every Banach space Y the fam ily M (v, Y) of v-measurable functions is the intersection of the fam ilies M (vt, Y).

(c) The fam ily M{v) of v-measurable sets is the intersection of the fam ilies M (vt).

P ro o f. Since vt{A) < v(A) for all te T and A e V we have from [8], Section 1, Theorem 1, that Ж(v), M (v, Y), M (v) are contained in П N (vt),

f} M (v t J Y) and П M{vt) respectively. T

T T

Lemma2. I f A e Q N (vt) and if there exists a set B e V such that A a В ,

thenA eN {v). T

P roo f. Take an e > 0, since £ v t {B )< oo there exists a finite set

T

8 с T such that £ vt(B) < e/2. Define the volume w on the prering V r\s

hy w = Since A eN {vt) for all te T we have from [8], Section 2, s

Theorem 1, that A eN (w ). Thus there exists a sequence of sets Gne V such that A c [ J Cn and £ w (G n) < e/2. From the properties of the

n n

prering V we may assume that the sets Gn are pair-wise disjoint and that Gn c В for all n. We have £ v (C n) = Z ( S vt(^ n ))+ 2 !( 2 vt(Gn))-

n n S n T \ S

But Z ( E v , ( C j ) = S ( Z M C n ) ) < Z щ ( В ) < Ф and Z i Z M C J

n T \ 8 T \ S n T \ S n S

= £ w (G n) < e/2. Therefore we have J]v(G n) < e.

n n

Now take a set In particular we have A e Ж(vtf) for

T

any fixed t0eT . Therefore from the definition of Ж (vtf) we have that there exists a sequence A ne V such that A <= U An. From the properties

П

of the prering V we can assume that the sets A n are pair-wise disjoint.

Let us set B n = A r\An. We have Bne П Ж{vt) and B n c An. Thus by

T

Lemma 2 the set B n is a v-null set. Since A = U -S» have АеЖ (и).

П

P ro o f of p a rt(b ). Denote by Va the family of sets which are unions of countable families of sets from the family V. If fe P) M (vt, Y), then

T

from the definition of the space M (vt, Y) we have that there exists a set B e Va such that f(oc) = 0 if x 4 B. Since B = U An for some sequence

П

of sets An e V and v (An) < oo for all n it is easy to show that there exists a subset TQ с T such that T0 is at most countable and r t(An) = 0 for all t(f. TQ and all n. Let us assume that T 0 = { L p2> •••}•

Denote by Y) the first Baire class generated by the family 8 ( V, Y) of simple functions, that is, the set of all functions / from X into the Banach space Y for which there exists a sequence of simple functions sne 8 (V, Y) such that the sequence of values sn(x) converges to the value f{x) for all x e X . 8i+1(Y ), the (ï-fl)-B aire class generated by 8 (V , Y) is the family of functions which consists of pointwise limits of sequences from the set 8{(Y) (i = 1, 2, ...). Notice that each of the Baire classes is linear and if ge 8k(B), f e 8k{Y), then gfe 8k(Y ). From [3], Theorem 7,

2 — Roczniki PTM — Prace M atem atyczne XVII.

324 W. M. B ogdanow icz and J. N. W elch

and the construction given in that theorem, it follows that a function / mapping X into Y belongs to the space M (v, Y) if and only if there exists a set De Vaôn N (v) such that the function cX\ Dfe $ 3( Y), where VaS denotes the family of sets which are intersections of countable families of sets from Va. It is not difficult to see that if fe M (v, Y) and f(x ) = 0 for x 4 Be Va, then we may assume that D <= B. Therefore for each tne T 0 there exists a set Ane Va$n N (v tn) such that A n с: В and cX\ Anfe $ 3(Y) (n — 1 , 2, ...). Set A — A n. Since A c A n for all n we have A e N(vt )

П

for all tne T0, but А с В and therefore A e N(vt) for all T0. Prom part (a) of this theorem we get A e N (v ). Define the functions gn — ex ^Enf, where En = J ^ n . .. n A ne Vaô. Notice the identity

9n ~ сх\ ги_1/+

for n > 1 from which using induction we get gne S 3(Y ) (n = 1 ,2 ,... ) . Por each xe X we have limgn(x) = cx^ A(x)f(x). This yields C j^ ^ e $4( Y)

П

c M (v, Y). But cX\ A(x)f(x) = f(x ) if x i A. Therefore f e M (v, Y) accord­

ing to [3], Theorem 1.

Prom [3] we have that the family M (v) of ^-measurable sets consists of all sets A e X satisfying cAe M (v, JR), where JR is the space of real num­

bers. Thus we have that part (c) of the theorem follows immediately from part (b) and the definition of the space M (v) of ^-measurable sets.

If v is any volume on the prering V, then in [3], Section 3, the set function mv is defined on the family M{v) of ^-measurable sets by the formula mv(A) = J c Adv for all A e M{v). In [3] it is proven that mv is the unique measure on the sigma-ring 71/ (v) being an extension of the volume v from the prering V.

Theorem 2. Let v, щ (te T) be volumes on the prering V such that v(A)

— for all A e V. Then we have

T

mv(A) = ^ m Vt(A) for all A e M (v ).

T

P ro o f. Let us define the set function m' on M (v) by the formula m '(A ) = f f m v (A) for all A e M (v ). From the previous theorem we have

t 1

M (v) = C\M(vt) and therefore m' is well defined. If A e V, then mVl(A)

T

— vt( A) for al I t e T yielding m' (A) = v(A ) for all A e V. If m' is a measure on M (v), then from [3], Section 3, Theorem 4, (7) we have that m '(A )

= mv(A) for all A e M (v). Let us now prove that m' is countably additive on M (v). Let A ne M (v ) be a sequence of pair-wise disjoint sets. Set A = (J An. Proceeding as in the proof of Lemma 1 we have

П

JP m '(A n) - £ ( ] [ mVt(An)) = = = m'( A )-

n n T T n T

Let (X, W, w) be a volume space. Denote by the family of all sets A <= X such that cAe L 1(w, B). Define the set function wc by means of the formula

wc(A) = j c Adw for all A e

We have that TF£, are just the measurable sets in AI(w) which have finite measure mw(A) = wc{A). For properties of the set function wc see [5].

The volume wc is an extension of the volume w. The volume wc is called the completion of the volume w.

Let t -> yt be a function from a set T into a Banach space Y. If ye Y же shall write у — yt if у is the limit in the space Y of the net z8 = £ y t,

t s

where 8 runs through all finite subsets of T ordered by inclusion.

Theorem 3. Let v ,v t (t e T⁾ be volumes on the prering ^V such that v(A) = fyvt{A) for all A e V. Then we have:

T

(a) П = {АсГ\Уе.г oo }<mdve(A) = if, (A) for all A eTJ.

T * T T

(b) For every Banach space Y we have 8 (V„, Y) c= P) 8 (VcVl, Y) and fsdv = £ fsdvt for all seS { V cv, Y). T

T

P ro o f. If A e VI c: M (v) = (~)M(vt), then mv(A) — vc(A) < oo.

T

Therefore Tm v (A) < oo and so vï(A ) — mv (A ) < oo for all te T . This

t 1 1

gives i f f \V% and vc(A) = ^ V t(A ). Now if A e <= f^ M (v t) = M (v)

t 1 t t 1 T

and ] ? v ct(A )< oo we have mv (A) = vct{A) for all te T and mv(A)

t 1

= ^ v n v (A) < oo. This gives us i e 7 J completing the proof of part (a).

t 1

P ro o f of p a r t (b). The first part of (b) follows from part (a) and the definition of the space 8 {V, Y) of simple functions generated by a prering V. Take the simple function s = cAy, where ye Y and A e Vcv.

From the continuity of scalar multiplication in the Banach space Y we have П

yvc{A) = J£yvt(A). If we take a simple function s = £ cA-yie®(Vv> ЗГ)

T i= i 1

we have

n n n

f sdv° = £ V b^iA ) = ' ] ? ( 2 ! У А Ш ) = У J sdtf.

i=1 г=1 T T i= 1 T

From [4], Section 1, Theorem 1, we have that the spaces L ^ v, Y) and L x{vc, Y) coincide and ffd v = ffdvc for all f e L 1(v, Y). Therefore we have fsd v = £ fsd vt for all Y).

T

For any volume v on the prering V denote by M (v, [0, oo]) the family of non-negative extended measurable functions as defined in [3],

■326 W. M. B ogdanow icz and J. N. W elch

p. 255. Denote by S +(V °,R ) the family of non-negative real-valued functions se /S(V%, R). In [3], Section 2, is defined the integral Jfd v for fe M (v, [0, oo]). It is not difficult to see that

j f d v = s u p jjs d r: se 8+(VI, R), s ^ f v-a. e.J.

Prom [3], Section 2, Theorem 3, we have that if fe M (v, Y), then

\f\eM(v, [0, oo]) and we have the relation f e L ^ v , Y) if and only if j\f\dv < oo.

Theorem 4. Let v ,v t (t e T⁾ be volumes on the prering V such that v(A ) = ]£vt(A) for all A e V . Then we have:

T

(a) The space M (v, [0, oo]) of extended non-negative v-measurable functions is the intersection of the spaces M (vt, [0, oo]),

(b) We have the equality / fdv = f fdvt->for all functions fe M (v, [0, oo]).

T

. P ro o f of p a r t (a). Prom [3], Section 4, Theorem 5, we have that a function/from the set X into the interval [0, oo] belongs to M (v, [0, oo]) if and only if there exists a set l e Va such that f(x ) = 0 if x 4 A and A n / _1(I)e M (v) for any interval of the form I = ( — 00, a]. Prom the identity M (v) = C\M (vt) we get part (a).

T

P ro o f of p a r t (b). Por any f e M (v, [0, 00]) it is not difficult to see that there exists a sequence sne 8+(V°, R) such that the sequence of values sn(x) increasingly converges to the value/(a?) for all x e X . Prom [3], Section 2, Theorem 3, (4), we have lim fsndv = Jfd v and lim fsndvt

— ffd v t for all te T . Since j s ndv = f sndvf (n = 1, 2, ...) to complete T

the proof of part (b) we need only show that lim 2 J s”dvi = 2 f f dvt’

Por each positive integer n there exists a set A ne V% such that sn(x) = 0 if x 4 A n. Since vc(An) < 00 there exists a subset T 0 <=. T satisfying T0 is at most countable and Vi(An) = 0 for all t 4 T0 and all n. Let us assume that T 0 = {tx, t2, ...}. Let us now show

и™ T !* .* % * = Y

i i

Let w be the volume on the prering V = {0, {j}: je J ] , where J is the set of positive integers and w {j} = 1 for all j e J . As we saw the sequence of numbers sn(x) increasingly converges to the value f(x ) for all x e X , the sequence of values j s ndvt increasingly converges to the value ffd v t for all te T and the sequence f s ndv increasingly converges to the value

ffd v . Therefore for the non-negative valued functions hn, h on J given by the formulas h (j) = £ c{i}(j) ffd v ti, hn(j) = ^ o {i}( j ) f s ndvf. for all

i i

je J and all n we have that the sequence of numbers hn(j) increasingly converges to h (j) for all j e J . But hn, he M (w, R) for all n and therefore from the Monotone Convergence theorem for extended real valued w- measurable functions (see Theorem 3, Section 2, [3]) we have lim f h ndu>

= fhdw. Let us notice that

f h ndw = JT J s ndvt. and fhdw = £ j

i i

thus part (b) is proven.

If v is any volume on the prering V and (Y, | |) is any Banach space,, then for each f e L ^ v , Y) we set ||/||„ = f\f\dv.

Theorem 6. Let v, vt (t e T ) be volumes on the prering V such that v{A)

= £ v t(A) for all A e V. Then we have:

T

(a) L x(v, Y) = \fe f i n i t e , Г): Z\\f\lvt < °°) and \\f\\v = 2\\f\\Vt for

all f e L x(v, Y), ^{t t t}

(b) f f d v = % ffd v t for all fe L x{v, Y).

T

P ro o f of p a r t (a). From [8], Section 1, Theorem 1, we get L x(v, Y) c Y) for every Banach space Y since the volume v dominates

T

the volume vt for every te T . If f e L x(v, Y) we have \f\e M (v, [0, oo]) and [|/||„ = f\f\dv< oo. Since M (v, [0, oo]) = (~\M{v, [0, oo]) and f\f\dv = £ f\ f\ d v t from Theorem 4, we get £\\f\L = ||/||„ < oo and

T T

therefore fe Ç^\Lx{vt, Y). Now let us take a function fe Q L x(vt, Y) such that ll/ll < oo. Since M (v, Y) = r\M {vt, Y) and L x(vt, Y) c M(vt, Y)

T T

we get f e M (v , Y) and therefore \ f\ eM (v,[0, oo]). From Theorem 4 we get that }\f\dv = 2}f\f\dvt < oo and therefore f e L ^ v , Y). Thus part (a) is proven. T

P ro o f of p a r t (b). Let v be any volume on the prering V; then a sequence of functions sn is called v-basic if there exists a sequence of simple functions hne 8 {V, Y) and a constant M > 0 such that sn = hx + + ... + h n and jlh ^ d v ^ Ж - * for i = 1 ,2 , ... From [1] we have that a function / mapping X into Y belongs to L x(v, Y) if and only if there exists a v-basic sequence sn such that the sequence of values sn(x) converges to the value f(cc) v-almost everywhere (v-a.e.). If se S (V , Y) is given by s ==c^12/i+--- +СлпУт then from [1] we have J|s|dv = lyxlviA j) + + ••• + \yn\v(An). Now if we take v to be the volume v = £ v t, then

T

since vt{A) < v(A) for all A e V we have that a v-basic sequence is also

328 W. M. B o gdanow icz and J. N. W elch

a tybasic sequence for all te T. Since N (v) = (~)N(vt) we have that if

t

sn is a v-basic sequence, for v = convergent -y-a.e. to /, then this

T

same sequence sn is ^-basic and convergent rr a.e. to the function / for all te T . This gives us limfs^dw = ffd v and lim/sndty = J fdvt for all

n 1 n

t e T. From Theorem 3 we have fsndv = f sndvt for all n. To complete

T

the proof of part (b) let us show

lim EJ J s«dvt = EJ j f dvt-

As in the proof of Theorem 4 we have that for each positive integer n there exists a set A ne F£ such that sn(æ) = 0 if a>4 An and therefore there exists a finite or countable subset T 0 a T such that vct (An) = 0 for all t 4 T0 and all n. Let us assume that T 0 = {t±, t2, ...}. We will now show

lim £ f s ndv,. = £ jf d v ,..

n i i

Let w be the volume defined as in the proof of Theorem 4. Consider the functions h, hn (n = 1, 2, ...) on J defined by

M l) = JE c{*№ J f d% ’ hnU) = JE C{i}Ü) f *nà \

i i

for all j € J . Using the fact that \\sn\\v = £ ||*»|L (n = 1 , 2 , . . . ) ||/[|„ = £ [|/||v

T T

and the Dominated Convergence Theorem we get hn, he L 1(w, F) {n = 1, 2, . . . ) , / Kdw = snd% (n = 1 ,2 , . ..) and / hdw = £ }fdvt.. To

i i

complete the proof of part (b) let us show limj h ndw = Jhdiv. We have П

I J hndw - j hdw I = I E J (sn “ /) d'% I < JE\\sn -/!!%• •

i г

But from part (а) ^||*п—/||г<> = IK,,—/IK Since sn is a v-basic sequence

i 1

convergent to / v-я. e. from [1], Lemma 4 we have lim||sn—/||„ — 0.

П

If v is any volume on a prering V, and F is a Banach space, then for functions f e Lp ( v, Y ) we set \\f\]vp = (f\f\pdv)llp for all numbers p satisfying 1 < p < oo. Since we will be considering different volumes, for functions f e L ^ iv , Y) let us denote the seminorm Ц/IK as defined in the introduction by ||/ЦЮ00.

Theorem 6. Let v, vt(te T) be volumes on the prering V such that v(A)

— ]£ vt(A) f or A e V. Then for any Banach space Y we have :

T

(a) L v(v, T ) = {/e П -М »,, Г): I'll/ll^,< °°) ««<* [l/!U = (^ 11/!!^)

T T T

for 1 < p < oo.

(b) L M{v, Y) = J/e П I'oo (vt, У): siip{||/|!eoo} < ooj and ||/1|воо

21 T

= Bupdi/^n}.

T

P roo f. The proof of this theorem is similar to the corresponding proofs of Theorems 6 and 5 in [8], Section 2.

2. Relations to Lebesgue, Bochner and Dunford approaches. In [13]

references are made to the relations that exist between the Bochner- Pettis, Dunford and Bogdanowicz’s approach to the theory of integration.

In the following we will discuss the principal papers, where such relations between the various approaches are treated and point out explicitly these relations. We will then be able to formulate the results of Section 1 in the classical setting.

From the results of [3] we have that the class of spaces of Lebesgue- Bochner summable functions generated by a sigma finite complete measure coincides with the class of spaces generated by a volume. Also in [3] the measurable functions corresponding to the approach to in­

tegration based on volumes are introduced and the classical relations between weak and strong measurability and summability (due to Pettis) are established. In fact, one can prove that if % , m2 are two complete measures, then for every Banach space Y we have that the space of Lebesgue-Bochner summable functions with values in Y generated by the measure m1 is equal to the space of corresponding summable functions generated by the measure m2 and the integrals J/dwT and jf d m 2 coincide if and only if the finite part of the measures coincide. A detailed discussion of similar relations between the classical Lebesgue integration theory and the theory of integration generated by a volume have been established in [7].

The transition from the theory of integration generated by volumes to the theory of integration of Dunford generated by a countably additive measure defined on a sigma algebra of sets on the basis of the results contained in [1] and [3] is the following. If m is a positive extended real­

valued measure on a sigma algebra as in Dunford and Schwartz [9], p. 144-155, and if v is the restriction of the measure m to the prering consisting of all sets of finite measure, then v is a volume in our sense and the Dunford construction of the space L\(m, Y) of Lebesgue-Bochner summable functions with values in the Banach space Y as defined on p. 119 of [9] yields the space L(v, Y) obtained by means of the construction in [1] and the Bochner integrals coincide on them.

It is also easy to see that the space L\(w, Y)generated by a complex­

valued measure w on a sigma algebra coincides with the space of such

330 W. M. B ogdanow icz and J. N. W elch

functions generated by its variation \w\ and thus one may assume that the measure generating the space of summable functions is positive.

In this case the trilinear integral f u 0(f, dw) developed in [1] is equal to the integral of Dunford ffd w , where the bilinear operator u0 is defined by the formula u 0{y, z) = zy for all у in the Banach space Y and all scal­

ars z.

Conversely, if v is any volume and m is the complete measure obtained from it in [3], then denote by M v the sigma algebra of all subsets A of X such that A n В yields a -y-summable set for every -y-summable set B.

Гог each set A in M v let m (A) = m (A) if A is ^-summable and m (A) = oo otherwise. One can show that m is a measure and that the space L\(m , Y) coincides with the space L(v, У) developed in [1] and the integrals are equal.

The approach to integration based on volumes yields a number of new results which were not noticed in constructions employing measures.

Гог example, we have that the class of operators obtained by means of the constructions in [1] and [3] contains as a proper subset the integral operator ffd w as defined in [9], p. 112, for functions / with values in the Banach space Y, where the complex valued measure w is defined on a sigma algebra. To see this take for example any countably additive complex valued set function q defined on a prering Г of a space X such that its variation |g| is infinite on X . Then the function q cannot be extended to a measure on the smallest sigma algebra containing V and thus the integral ffd q constructed in [9] cannot be defined, where as by means of the construction in [1] it can.

As is well known, the result of Kakutani [12] on abstract i-spaces shows the importance of the linear-lattice structure in the theory of integration. Let us also notice that the notion of a prering is a natural one in this context for the following reason. If F is any family of subsets of a space X and S ( V ,B ) the family of real-valued simple functions s = г гсА1+ ... + ткcA]c, where A { is a finite family of pair-wise disjoint sets from F, then F is a prefing if and only if $ (F , B) is a linear lattice.

In fact, even the requirement that $ (F , B ) is a group under addition is equivalent to F being a prering. (See [6], p. 206.)

If v is any volume on the prering F, then as in Section 1 let us denote by mv the measure defined on the family M (v) of ^-measurable sets by the formula

mv(A) = J e Adv for all sets A e M (v),

and by vc the volume on the prering Fc = {Ae M (v): mv(A) < oo} given by the formula ve(A) = mv{A) for all sets A eV c.

Let m be a non-negative extended real valued measure on a sigma-ring M of a space X . The triple (X , M, m) will be called a measure space.

Denote by Lm(B ) the space of finite real-valued Lebesgue summable functions without identification of functions equal m-a. e. (see [7], p. 281-289). Let Qm denote the Lebesgue completion of the measure m (see [10], p. 55 or [7], p. 291).

For any family ТУ of subsets of the space X let a(W ) be the smallest sigma-ring of subsets of X containing the family ТУ.

Lemma 1. Let (X, M , m) be a measure space. I f W is a prering of subsets of X such that ТУ <= V <= cr(W"), where V — ( l e M : m(A) < oo}, then for the volume w, defined on the prering W by the formula w(A) = m (A) for all A e W , we have L x(w, B) — La (B) and jf d w = j fdQ m for all f e L x(w, B).

P ro o f. Since F с У c or (ТУ) we have o'(ТУ) = a(V ). Define the volume v on the prering V by v(A) = m(A) for all A e V. By [7], p. 294, the volumes v and w have unique extensions to measures mx and m%

respectively on a(V ) = a{W) given explicitly by mx{A) = mv{A) and m2(A) = mw(A) for all A e a(W ). Since the volume v is an extension of the volume w we have mw(A) = mv(A) for all sets A e <7(ТУ). If A e V, then mw(A) = mv(A) < oo therefore J.eT yc and v(A) — wc(A). Thus wc is an extension of the volume v. Since the volume v is an extension of the volume w we also have that the volume vc is an extension of the volume w. From [4], Section 1, Theorem 2, we get vc = wc, and from [4], Section 1, Theorems 2 and 1, we have L x(v, B) = L x(w, B) and jf d w

= jf d v for all f e L x(v ,B ). But from [7], Theorem 2, p. 291, we have LQm{B) = L x(v, B) and jf d Q m = Jfd v for all f e L x(v ,B ). Therefore we have

Lam(B) = L x(w, B) and Jfd Q m = J f d w for all fe L x(w, B ).

For convenience of notation in the remainder of this section, let us adopt the convention that if (X, M ,m ) is a measure space and (X, M t, mt) a family of measure spaces for t in an index set T, then we set Q = Qm and Qt = Qmt for all te T, where Qm, Qmt denote the Lebesgue completion of the measures m and mt.

Theorem 1. Let (X, M , m) be a fixed measure space and (X, M t , mt) a fam ily of measure spaces, where the parameter t changes in an index set T.

Assume that W is a prering of subsets of the space X such that ТУ с V —

= {Ae M : m(A) < oo} and m (A) = £ m ((A) for all A e W . I f the fam ily

T

сг(ТУ) contains each of the fam ilies V, Vt = {AeM t: mt(A) < oo} (te T ), then

- M - R ) = { / с П

LatW- L I

dQt <

jfd (i = L I f d£}‘ f0*

al1 f ' L° •

and

332 W. M. B ogdanow ciz and J. N. Welch.

P ro o f. Prom Lemma 1 for the volumes w ,w t(te T ) defined on TP byw(-A) = m (A ),w t(A) = mt(A) for all .A e Ж we have L n (B) = L 1(w, B),

I fdQ = jf d w for all fe L x(w, B) and L Qt(R) = L x{wt1 B), J fdQt = f fdwt for all fe L ^ W fjB ) and all te T . The theorem now follows immediately from Theorem 6, Section 1.

Theorem 2. Let X = В ^bethe space of real numbers, and mt be a fam ily of non-negative Borel-regular measures on the fam ily В of Borel sets of В for t in a parameter set T. For each te T let gt be the left continuous increasing real-valued function on В satisfying gt{0) = 0 and mt [a, b) = gt{b) — gt(a) for all intervals [a , b) with a, be В and a < b. Define the function g on В by д(я) = ^У Л Х) f or °М X^B . I f g (x) < oo for all x e B and В is the set

T

of discontinuities of the function g, then for the Borel-regular measure m on В satisfying m [a ,b ) = g (b ) — g(a) for all a ,b e R \ D , a < b, we have:

(i) The Lebesgue completion B m of the fam ily В of Borel sets relative to the measure m is equal to the intersection over all te T of the Lebesgue completions В щ of the fam ily В relative to the measures mt and Q(A) =.

= S Q tW f or al1 AeBm -

T

(ii) The space L Q(B) = | / < П ^ ,М ; < °°| and ffdQ

= S f f d Q , f o r a U f 'L 0(R). T T

T

P ro o f. Prom [11], p. 332, we have that the measures mt are Borel regular if and only if mt [a, b) < oo for all intervals [a, b). Since ^(0) = 0 for all < € T it follows that the measure m defined on the family В of Borel sets hy the formula

m (A) = ^ m t(A) for all A e B

T

is Borel regular if and only if J£gt(x) < oo for all x e B , hut hy assumption

T

g(x) < со for all x e B . Let D be the set of discontinuities of the function g and let W be the prering

W = { [a,b ): a, be R \ D , a < b}.

Since В is countable it follows that the smallest sigma-ring of subsets of В containing TP is the family of Borel sets B. It is easy to see that the function v defined on the prering W by the formula v [a, b) = g{b) — g{a) for all [a, b)eW is a volume on W and v(A) = ^ v t(A) for all A e W,

T

where vt is the volume on W given by vt(A) = mt(A) for all A e TP. If m denotes the unique measure defined on the Borel sets В which is an extension of the volume v, then we have m = m since m(A) = £ v t(A)

T

= v(A) for all A e TP. Since we noted that or(TP) = B, from Theorem 1 of this Section we get part (ii). Part (i) follows from Theorems 1 and 2 of Section 1, and from [7] Theorem 1, p. 294.

Let (X, M, m) be a measure space with M a sigma-algebra. Set V = {Ae M : m(A) < oo}, and define the volume v on V by v(A) = m(A) for all i e 7 . As we noted in the first part of Section 2 the space L\(m, Y) of Lebesgue-Boehner summable functions with values in the Banach space Y obtained by means of the construction in [9] is equal to the space L x(v, Y) obtained in [1] and the integrals ffdrn and Jfdv coincide.

We now easily get:

Theorem 3. Let ⁽X, M , m) be a fixed measure space and (X, Mt, ^{mt )} a fam ily of measure spaces with M, M t sigma-algebras for all t in an index set T. Assume that W is a prering of subsets of the space X such that W cz V

— {Ae M : m(A) < oo} and m (A) = JFm*(A) for all A e W . I f the sigma-

T

ring a(W ) contains each of the corresponding fam ilies V, Vt = {A e M t : mt(A) < oo} (t e T ), then

L\(m, Y) = {/с П £!(»%, Г): У f ooj

1 T у ’

and

j f d m = fdm , for all /* L\(m, Y ).

T

P roo f. Define the volumes w, wt (t e T ) on the prering W by w(A)

= m(A), wt(A) = mt(A) for all A e W . From the proof of Lemma 1 of this section we have for the volumes v ,v t (teT ) defined on V, Vt by v(A) = m (A) for all A e V, and vt(A) — mt(A) for all A e V t respective­

ly, that the completions agree, that is, wc = vc, and (wt)c = (vt)c (teT ).

From [4], Section 1, Theorems 1 and 2, we get, for every Banach space Y , L x(w, Y) = L i(v, Y) and jf d w = Jfd v for all f e L x(w, Y) and the same for wt and vt for all te T . Thus from Theorem 5 of Section 1 the theorem follows.

References

[1] W. M. B ogdanow icz, A gen eralizatio n of the L eb esgu e-B och n er-S tieltjes in te g ral an d a new approach to the theory of in tegratio n , Proc. Nat. Acad. Sci. U.S.A.

53 (1965), p. 492-498.

[2] — A n approach to the theory of L p spaces of Lebesgue-Boehner sum m able fun ctio n s an d generalized L eb esgu e-B o ch n er-S tieltjes in teg ral, Bull. Acad. Polon. Sci., Ser. sci. math., astr. et phys. 13 (1965), p. 790-800.

[3] — A n ap p ro ach to the theory of Lebesgue-B oehner m easurable fun ctio n s an d to the theory of m easure, Math. Ann. 164 (1966), p. 251-269.

[4] — On volum es gen erating the sam e L ebesgue-B oehner in tegratio n , Proc. Nat. Acad.

Sci. U.S.A. 56 (1966), p. 1399-1405.

[5] — E xistence an d uniqueness of extensions of volumes an d the operation of completion of a volume 1 , Proc. Japan. Acad. 42 (1966), p. 571—576.

[6] — Lebesgue-B oehner type spaces of g ro u p -v alu ed sum m able fun ctio n s an d abstract in teg rals, Proceeding of the Symposium in Analysis, Queen’s Papers in Pure and Applied Mathematics 10 (1967), p. 205-208.

334 W. M. B ogdanow icz and J. N. W elch

[7] — B elatio n s between the Lebesgue in te g ra l generated by a m easure a n d the in te g ra l generated by volum e, Comm. Math. 12 (1969), p. 277-299.

[8] — and J . N. W elch , In teg ratio n generated by a volume being the sum of volumes w ith a n a p p lic atio n to sp ectral in teg rals, Bull. Acad. Polon. Sci., Ser. sci. math., astr. et phys. 19 (1971), p. 719-726.

[9] N. D u n fo rd and J. S c h w a rtz , L in e a r operators I , Interscience Publishers, Inc., N. Y. (1958).

[10] P. H alm os, M e asu re theory, New York 1950.

[11] E. H ew itt and K. S tro m b e rg , B e a l a n d abstract a n a ly s is , New York 1965.

[12] S. K a k u ta n i, Concrete representation of abstract L -spaces an d the m ean ergodic theorem, Ann. of Math. 42 (1941), p. 523-537.

[13] E. J. M cShane, A B iem an n -typ e in te g ra l that in clu d es L eb esgu e-S tieltjes, Bochner an d stochastic in te g ra ls, Mem. Amer. Math. Soc. 88 (1969).

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