Goldberg, Robert R.

Simon, A. B.

English

Goldberg Robert R.

Simon A. B.

University of Szeged

Characterization of some classes of measures By R. R. GOLDBERG and A. B. SIMON in Evanston (Illinois, U. S. A.)1) I. Introduction Let M(G) be the set of (bounded regular Borel) measures n on a locally compact abelian group G. The following theorem is a well-known characterization of those measures which are absolutely continuous (with respect to the Haar measure of G), given iu terms of the translates ¡ix (where nx(E)=n(E — x)). T h e o r e m A. Let nd M (G). Then n is absolutely continuous if and only if ll^jc-^11 a s 0-For a proof see ([5], p. 230). The norm in the statement of Theorem A is the usual measure norm ( = total variation). In this paper we introduce two other norms for M(G). Using them we give 1) a characterization- of the measures in M{G) whose Fourier—Stieltjes transforms vanish at infinity, and 2) a characterization of the continuous ( = non-atomic) measures in M(G). In each case the necessary and sufficient condition is similar to that in Theorem A — namely, that as ¡xx must approach /( in a suitable norm. In case 2) we must restrict ourselves to metrizable groups. n . Characterization of measures whose Fourier—Stieltjes transforms vanish at infinity Let r denote the character group of G. If n £ M(G) let |M| r = sup \fi(y)\ ver where fL is the Fourier—Stieltjes transform of fi — that is fi(y) ~ J (y, t)dn(t). Since G p. is determined by the values of fi on T [5], we have ||^||r = 0 if and only if fi is the 0 measure. The other conditions that || - | | r be a norm are readily verified. Here is our characterization. T h e o r e m B. Let n£M(G). Then fl vanishes at infinity ifand only if H/^ —ju||r —0 as jc—0. ') Research supported by the National Science Foundation Grant GP-3930. 158 R. R. Goldberg — A, B. Simon P r o o f . Suppose first that p. vanishes at infinity. Since A . G O = ¡ ( y T t j d n J t ) = J faTjd(x(t-x) = f (y,t + x)d/i(/) = (yTx)fi(y), G G G we have yil ytr Since, by assumption, u vanishes at infinity,'given1 £ > 0 there Exists a compact KczT such that | £ ( y ) | < y i f y£F — K. Moreover, the set U of all x in G such that |(y, x) —1| <e/||ju|| r for all y£K is a neighborhood of 0 in G. If y£K we thus have ( 2 ) l ( 7 , * ) - l | - | / K v ) l < ( « / I N I r ) - I N I i - = e ( * € C / ) , while if y ^ r ^ - K we have . . (3) \ ( y , x ) - l \ . \ ( l ( y ) \ ^ 2 . j = £ ( . v e C ) . From (1), (2), (3) it follows that \\{ix-ti\\r<e if jc6 U. That i s j | ^ x - / t | | - 0 as x - 0 . This proves half the theorem. To prove the other half we need a lemma (see [1]). L e m m a . Let Ube any neighborhoodofO in the locally compact abelian group G. Then there exists a compact subset K of T (the character group of G) such that for tiny y^T — K there exists x£U with Re (y, x) SO. ¡Now suppose \\px — n\\r~*0 as x —0. We must show that fi vanishes at infinity. Given e > 0 choose a neighborhood U of 0 in G such that \\[ix —p\\r<e (x6 U). Then. (4) | ( y , x ) - l | . | £ ( y ) | < 8 ( y € r ; x 6 £ 7 ) . For this U choose KaT according to the lemma. Then if y£F — K there exists x £ U with Re (y, x) ^ 0 so that |(y, x) — 11 > 1. Using this x in (4) we have |/i(y)| < e-if y£T — K. This completes the proof. III. Characterization of continuous measures If G is a non-discrete metrizable group then its character group T is tr-compact. In F there is a sequence of open subsets An with compact closure satisfying; Atc:A2cz ..., lim m(A„) = and such that (5) 1 M«) = !zd6f/(y)dy A„ exists for all almost periodic functions / on f and is equal to the mean value of / ([2]). Here m(An) means the Haar measure of An. H E W I T T and STROMBERG ([3]) have shown that the limit in (5) will exist for many other functions as well, and, in fact they proved Some classes of measures 159-L e m m a . Let ¡i^M[G). Then ¡xis a continuous measure if and only if M(\fi\)=0. We now define our second norm. If fi £ M(G) where G is metrizable, let • JT(n) = s u p — i — \fi(y)\ dy. n m{A,.)J An Our second main result is T h e o r e m C. Let G be a metrizable locally compact abelian group and let' u f M(G). Then n is a continuous measure if and only if jV(hx~h)-*0 as jc->-0. P r o o f . First suppose that /t is continuous. Then by the lemma we have M(|AI)= lim ——r / \fL(y)\dy = Q>. Hence, given 8 > 0 there exists N such that n->~ m(A„) J A„ (6) - J j j J d y <= 8/2 (nSN). AN Moreover, the set U of x in G such that |(y, x) — 11 < e/||/(||r for all y € AN is a neigh-borhood of 0 in G. Thus, if n=s N we have from (6) An An Also, if n^N then AnQAN and so, if x£U 1 m(A ^ J \ ( y , x ) - 11 • |/i(y)| dy == Mr • J \ ( y , x ) - l \ d y * A„ A„ • S MAelMr). Thus, if x 6 V, or or m(A„ s u p / - 11 • №(V)l DY = £> n m(A„)J An ^ f e - j " ) = e Thus, J r ( f i x - f i ) - ^ 0 as 0. This proves half the theorem. Now suppose that J f ( i i x — n)-» 0 as x—0. We must prove that /x is continuous. Given e > 0 choose a symmetric neighborhood U of 0 in G such that (7) jr(fix-p)^e (x£U). 160 R. R. Goldberg — A. B. Simon Let <p be the function on G defined by q>(t) = llm(U) (t£U), <p(t) = 0 (teG-U). (We are now denoting the Haar measure on G, as well as that on r , by m. We may clearly assume that m(U) -< Then <p, the Fourier transform of q>, is real-valued (since U is symmetric) and <p vanishes at infinity by the Riemann—Lebesgue theorem. Thus, for some compact i c f , 0(?) = i if y£T — K. That is, MY) = J(y, X)<p(x)dx = J(y, x)dx S J (yer-K). Hence ,x)]dxm-^ (yer-K), T u ) f l l - ( 7 " m(U) J Li V " ' V J 2 u and so (8) (ytr-JC). v Now from (7) we have for any n ^ AN If we multiply by ——— , integrate over U, and invert the order of integration we m(U) obtain , , . . „ „ • 11 dx^ s. m(A, A,, AN Then certainly k ) f i m d y - ^ u ) f l ( y > x ) -A„ AN ' A„-K V But if y£An-K then, by (8), J\(y,x)~\\dx^. Hence A„-K Moreover, it is certainly true for large n that Some classes of measures 161 since m(A„)-~ °° as n — °°. Hence ¿j/ An for large n, which proves that M(\fi\)= lim . . . I \fi(y)\dy =0 . By the lemma, J An fi is continuous and the proof is complete. IV. Remark on another class of measures We have made an attempt at another result along the lines of Theorems A, B, C. Let A be the maximal ideal space of the measure algebra M(G). That is, A is the space of continuous complex-valued homomorphisms h on M{G). If for fi € M(G) we define | | / j | |d= sup \h(fi)\ then, since h(p.) = ß(h) where fi is the Gelfand transform of n, U ^ is the spectral norm of ¡i[4: p. 76]. It is now natural to ask: For what fi is it true that \\fix—fi\\A —0 as x—0? For each x € G let <jx be the unit mass concentrated at x. For h£A the function Xh defined by X„(x)=h(ax) ( x e G ) is easily seen to be a group character of G. However, need not be continuous. If we could answer positively a certain question about these %h we could give a characterization of the kernel of the hull of Ll(G) in M(G) — the set of all ¡x £ M(G) such that fi(h) — 0 for all h^A—.G. The question whose answer we are unable to establish is this: Are the h for which Xh ¡ s discontinuous dense in A—Gl If the answer to this question is yes then we can easily establish the following: Let ß £ M{G). Then n is in the kernel of the hull of Li(G) if and only if — as x—0. References [1] R. R. GOLDBERG and A. B. SIMON, The Riemann —Lebesgue theorem on groups, Acta Sei. Math., 27 (1966), 35—40. [2] E. H E W I T T and K. Ross, Abstract Harmonic Analysis (Berlin, 1963). [3] E . H E W I T T and K . R . STROMBERG, A remark on Fourier—Stieltjes transforms, An. Acad. Brasil. Ci., 3 4 ( 1 9 6 2 ) , 1 7 5 — 1 8 0 . [4] L . LOOMIS, Introduction to Abstract Harmonic Analysis (New York, 1 9 5 3 ) . 15] W . R U D I N , Measure algebras on abelian groups, Bull. Amer. Math. Soc., 6 5 ( 1 9 5 9 ) , 2 2 7 — 2 4 7 . NORTHWESTERN UNIVERSITY (Received January II, 1966) 

Characterization of some classes of measures

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Characterization of some classes of measures

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