Perfect Subset Research Articles

The Hausdorff measure of the intersection of sets of positive Lebesgue measure

Erdos, Kestelman and Rogers [1[ showed that, if A1, A2,… is any sequence of Lebesgue measurable subsets of the unit interval [0, 1] each of Lebesgue measure at least η > 0, then there is a subsequence {Ani} (i = 1, 2,…) such that the intersection contains a perfect subset (and is therefore of power ). They asked for what Hausdorff measure functions φ(t) is it possible to choose the subsequence to make the intersection set ∩Ani of positive φ-measure. In the present note we show that the strongest possible result in this direction is true. This is given by the following theorem.

A remark on universal sigma integrability

Our purpose is to answer the question of Leader [1, p. 2341, by producing an example of a sigma algebra a of subsets of a set X and a universally a-integrable function f (i.e., ffdu exists for every countably additive set function u on a) which is not measurable with respect to a. We shall show that the sigma algebra a of Borel subsets of the interval [0, 1] admits nonmeasurable universally a-integrable functions (R. E. Zink suggested that this Borel algebra might yield an example). Lusin has shown [2] that, subject to the continuum hypothesis, there exists an uncountable subset V of X = [0, 1] such that each perfect nowhere dense subset G of X contains at most an enumerable set of points of V. Let H be a subset of V that is not a Borel set. We shall now show that the characteristic function x(H) of H is universally sigma integrable. To this end suppose u is a countably additive set function on a-. Then [5] u = c +j where c is continuous and j= Ijk where jk is a two valued jump function, the range of k is at most enumerable, and the variation of j is the sum of the variations of the jk'S; moreover, since the variation of u is the variation of j plus the variation c of c, each of j, c, and c is countably additive on a. Corresponding to each jk there is [4, Theorem 27.1] a point Xk such that jk(E) =jk(En [Xk ]) for every Borel subset E of X. Hence fX(H)dj= Ejk(HCn[Xk]) and it suffices to show that there exists a Borel subset B of X such that HCB and c(B) = 0 (then fX(H)dc= 0). There exists [3 ] a set K = U i Fi, F1 closed for i > 1, of the first category in X and a set L such that c(L) = 0 and KUL = X. For each positive integer i there exists a perfect set Gi and an at most enumerable set Hi such that Fi = Gi U Hi. Thus H n K = HC(Ui Fi) = Hf (Ui (GiUkHi)) = Ui (Hrl (GiU-Hi)) CUi((HnGi) UHi) is at most enumerable since every HrlGi (Fi is a closed subset of X and K is of the first category in X; hence Fi is nowhere dense in X which implies that G, is a perfect nowhere dense subset of X) and every H, is at most enumerable. Since e is countably additive on o and c vanishes on one point sets, c vanishes on countable sets. Hence H=Hr)(KUL) C(HnK)UL and c(HnK)UL) <?(HnK) +Z(L) =0.

Banach algebras with scattered structure spaces

1. For a commutative semisimple Banach algebra B, let 9lZB denote its space of nonzero multiplicative linear functionals, and x->)x its Gelfand representation. A well-known theorem of Silov [1; 15 ] shows that for any compactopen subset U of 9)fB there is an x in B with xe the characteristic function 4u of U. An immediate consequence is the fact that B is regular if 9)lB iS totally disconnected. The present note is devoted to a similar application of gilov's result which has apparently escaped notice. Normally when A is a subalgebra of B (closed or not), 9MA may contain functionals other than those provided by the restrictions of the elements of 9)B. But at least when A is closed the Silov boundary dA of A is produced by the elements of dB. In any case, if we assume aA is produced by aB, and that dB is scattered (i.e., contains no nonvoid perfect subset), then Silov's theorem shows not only that B is regular but indeed that A is regular as well so that all of 9J1A = 9A arises from 9B = 9B. When d9B is discrete the same is true of dA, and we can trivially identify the smallest hull-less ideal jA( ?? ) of A; it is precisely the span of the idempotents in A. Consequently A is tauberian if and only if it is the closed span of its idempotents. As an application we can easily determine all closed tauberian subalgebras of L1(G) and L2(G) when G is a compact abelian group. In the L2 case every closed subalgebra A is tauberian, and is determined by just the sets of constancy of the Fourier transforms A'; the same prescription applies to the tauberian closed subalgebras A of L1(G), but whether nontauberian subalgebras exist seems to be a difficult problem. Borrowing from the L2 case we can easily see that A is tauberian if (and of course only if) A nL2 is dense in A. (Some application can also be made to closed commutative semisimple subalgebras of L1(G) and L2(G) when G is compact nonabelian, cf. ?4.) The notation used below is essentially standard, as in [9], with the exception of our use of scattered, as defined above. We denote the hull of an ideal I by hI, and by kF the kernel of a subset F of 9)A, while jA( CO) is the set of a in A for which a has compact support. A is called tauberian when every hull-less ideal is dense in A; when A is regular this amounts to the density of jA ( oo) [9]. All algebras will be assumed commutative semisimple. Although ^ may be used for any Gelfand representation which arises, it will always be clear from the context which is intended. Finally, Aa will be used to

Spaces of continuous functions (III) (Spaces C(Ω) for Ω without perfect subsets)

Spaces of continuous functions (III) (Spaces C(Ω) for Ω without perfect subsets)

Continuous Functions on Compact Spaces Without Perfect Subsets

Continuous Functions on Compact Spaces Without Perfect Subsets

Continuous functions on compact spaces without perfect subsets

Continuous functions on compact spaces without perfect subsets

On the Distribution of Successive Images in the Poincare Transformation Problem of a Circle Into Itself

We consider the points on a circle C of radius 1/ (27r) and denote by x the distance, measured in a certain sense along the circumference, from a fixed point 0 on C to a variable point P. Since the circumference is of unit length, we agree that two values of x which differ by an integer shall correspond to the same point P on C. An equation of the form x f(x), where f(x) is continuous, monotonic and nowhere constant and satisfies the functional relation f (x + 1)= f (x) + 1, obviously defines a one-to-one order preserving transformation T of the points of C into themselves. Let us take x0 arbitrarily and let x, = f (xnw1), (n =1, 2, ). Then it is well known that xn an + 0+(1) as n -> oo, where a is a number independent of n or x,. It is also well known that a necessary and sufficient condition that a be irrational is that neither T nor any of its iterates have a fixed point on C. We shall assume that such is the case. The set S of the cluster points of the sequence {xn} is then independent of x0 and is either the whole eircumference C or a nowhere dense perfect subset of C. If S = C, the transformation x~ = f(x) may be transformed by a oneto-one order preserving transformation x= G(x) [t G(x 1) + 1] of C into a rotation of C through the irrational angle 2%ra, so that G{f[G-1 (x)] } x + c, where G-1 denotes the inverse of G. A similar result holds in the case S 7& C by introducing a proper convention regarding the asymptotically empty set S C. t Let + (x) be an arbitrary continuous periodic function with the period 1.

A connected and connected im kleinen point set which contains no perfect subset

A connected and connected im kleinen point set which contains no perfect subset

On trigonometrical series whose cesàro partial summations oscillate finitely

§ 1. Riemann’s remarkable theorem, which, in the extended form given to it by Lebesgue, by virtue of the use of the concept of generalised integration, asserts that a trigonometrical series is a Fourier series if it converges every­ where, except at a reducible set of points, to a function which is summable and has a value, or values, everywhere finite, has been discussed, and still further extended, by a relatively large number of writers. The object of the present communication is to state and prove certain results which include all those at present known. They are as follows :— I. If the upper and lower functions of the succession of the Cesàro partial summations, index not greater than unity, of a trigonometrical series ∞ ∑ r =1 (a r cos rx + b r sin rx ) ≡ ∞ ∑ r =1 A r , which is such that a n ⟶0 and b n ⟶0 as n ⟶ ∞, are summable, and everywhere finite except possibly at a set of points which, contains no perfect sub-set, then the series is a Fourier series. II. If the upper and lower functions of the succession of the Cesàro partial summations, index k (0 ⦤k &lt;1), of a trigonometrical series ∞ ∑ r = 1 and everywhere finite, then the series is a Fourier series.