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An example will be given of a compact linear set K which is of zero or non-σ-finite measure for every translation-invariant Borel measure. This answers a question asked me by C. Dellacherie.

Introduction. We recall (see Morgan Ward (3), Birkhoff (1), p. 49) that a closure operator on a complete lattice ℒ is a mapping f from ℒ into ℒ satisfying the conditionsfor all X, Y ∈ ℒ. The set of all closure operators on ℒ is itself a complete lattice with respect to the partial ordering in which f1 ≤ f2 if and only if f1(X) ≥ f2(X) for all X ∈ ℒ for every subset ℱ of and every element X ∈ ℒ we have.

The contribution “Mathematics of the pop charts” by E. de St. Q. Isaacson (The Mathematical Gazette, 51 (1967), 282-287) prompts me to publish a result found in 1959 during research on the same subject.

Goldman (4) conjectured that if Z is a linear set having the property that for every (Lebesgue) measurable real function f the set f−1[Z] is a measurable set, then Z must be a Borel set. I pointed out (2) that any analytic non-Borel set provides a counterexample, and Eggleston(3) showed that a set can have the property but be neither analytic nor even an analytic complement, for example, any Luzin set. As Eggleston mentions, in the construction of Luzin sets the continuum hypothesis is assumed (compare Sierpiński(6), Chapter II), and the question arises whether it can be dispensed with in his theorem. We shall show that a non-analytic set having Goldman's property can be constructed with the help of the axiom of choice alone, without the continuum hypothesis; the problem for analytic complements remains open. We shall also generalize one of Eggleston's intermediate results.

A. J . Goldm an (On measurable se ts and fun c tions, J. Res. NBS 698 (Math. and Math . Phys .) Nos. I and 2, 99100 (1 96.5)) conjectured tha t the Borel se ts are c harac te ri zed by the ir propert y of having meas urable inverse images unde r a ll Le besgue measura ble func tions : he re it is po inted out tha t the e xi s t e nce of analyti c non· Borel sets refut es th is and a re la ted conjeclure. Also an error in Goldm an 's T heore m :2 is correc ted.

Cantor's Theorem is generalized to a theorem on partially ordered sets. We shall show that every monotone mapping of a complete lattice into itself has a point of left continuity and a point of right continu- ity. From this result we derive an extension of a theorem of Gleason and Dilworth (2) which in turn can be regarded as a generalization of the classical theorem of Cantor stating that the cardinal of a set is less than the cardinal of its power-set. As a corollary it follows that if E and F are partially ordered sets then the cardinal power FE is not a homomorphic image of E unless \F\ =1. This result answers a question of F. W. Lawvere which provided the stimulus for our investigation. 1. A continuity theorem for complete lattices. If E and F are partially ordered sets then a mapping (a) = V (a) = A d>(x).