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1. THE problem treated here is an example of a wide class of problems involving a space E which has (i) a binary law of composition, e.g. addition, group multiplication; (ii) a real-valued monotonic set-function /x.Let the law of composition be denoted by o. Then, HA,B are two subsets of E, we define AoB = {xoy\zeA,ye B}. Our problem is: given fx{A), fi (B), what can be said about the value of fi{A oB)?Well-known results of this type are the (a-f/3)-theorem, first proved by H. B. Mann (10), and the analogous Cauchy-t)avenport theorem on finite cyclic groups (4, 5). In the first of these ft is the density of the set, in the second it is simply the number of elements.In the present paper we deal with Euclidean n-space, /* being the Lebesgue measure, and the composition law being vector addition. Our inequality was first proved by Brunn (3) for convex sets, and conditions for equality to hold were added by Minkowski (11). In 1&35 Lusternik (9) generalized the result to arbitrary measurable sets and stated conditions for equality which, as we shall see, are correct, though Lusternik's proof of them is defective. A certain amount of recent literature has appeared applying the result to the isoperimetric problem, and, in particular, mention should be made of (13), E. Schmidt's work generalizing certain restricted forms of the result to non-Euclidean geometries. See also the paper by Hadwiger (6).Lusternik's proof is open to the general criticism that, although he states the result for arbitrary measurable sets, he is inclined to assume that these sets have special properties, e.g. that their intersections with linear subspaces of E n are measurable in the lower dimension. This is false, in general, but the difficulty can easily be surmounted by assuming the sets to be J^-sets (as we do), and then deducing the more general result from the principle that every measurable set contains an F a -Bot of the same measure. Thus Lusternik's proof of the inequality is essentially valid.There is a more serious fault in his proof of the conditions for equality. At (9), p. 57,11. 30 sqq., he is considering the situation of a hyperplane M o . Ifp e M o , he denotes by L^ the line through p perpendicular to M Q . A and B are assumed to be two sets of finite positive measure, and it is supposed

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The introduction of a mathematics student to formal integration theory usually follows the lines laid down by Riemann and Darboux. Later a change of ideas is necessary if he tackles Lebesgue's more powerful theory, and connections between the two are laboriously constructed. On the other hand, the commonest method of evaluating an integral is through the operation inverse to differentiation (the indefinite integral taken between limits). We refer to this as the calculus integral; few realize that this can succeed in cases where even the Lebesgue integral does not exist, let alone the Riemann one. An example is given later.