Measure, Lebesgue Integrals, and Hilbert Space by A N Kolmogorov PDF

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By A N Kolmogorov

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3) The system of all finite subsets of an arbitrary set A forms a ring of sets. This ring is then and only then an algebra if the set A itself is finite. a unit. From the definition of a ring of sets there immediately follows Theorem 1. The intersection m = 0 ma of any set of rings is also a ring. Let us establish the following simple result which will however be important in the subsequent work. Theorem 2. For any non-empty system of sets ~ there exists one and only one ring m(~), containing ~ and contained in an arbitrary ring mwhich contains ~.

It is easily seen that all the sets n are pairwise nonintersecting, and that they add up to the whole circular arc C. If the set were measurable, then the sets n, congruent to it, would also be measurable. L() > 0. Thus the set (and hence also each n) is non-measurable. 2. SYSTEMS OF SETS 19 2. Systems of Sets Before proceeding to the general theory of sets we shall first give some information concerning systems of sets, which supplements the elements of set theory discussed in Chapter I of Volume I.

K=l It is easy to see that the quantity ~(A), given by equation (2), does not depend on the selection of the decomposition (1). Indeed, let us consider the two decompositions A = UB; = lJ C;, i=l i=l Since all intersections B; n Cj belong to Sm, we have, because of the additivity of measures, m n L: m(B;) = L: L: m(B; n C;) i=l i=l L: m(C;), 4. CONTINUATIONS OF JORDAN MEASURES 29 which was to be proved. L( A), given by equation (2) is non-negative and additive is obvious. L( A) of the measure m( A) is shown.

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Measure, Lebesgue Integrals, and Hilbert Space by A N Kolmogorov

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