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Products of Two Borel Measures by Roy A TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 269, Number 2, February 1982 PRODUCTS OF TWO BOREL MEASURES BY ROY A. JOHNSON Abstract. Let ¡j, and v be finite Borel measures on Hausdorff spaces X and Y, respectively, and suppose product measures ¡i X x v and p. X2 v are defined on the Borel sets of X X Y by integrating vertical and horizontal cross-section measure, respectively. Sufficient conditions are given so that ¡i x¡ v = p x2v and so that the usual product measure u X v can be extended to a Borel measure on X X Y by means of completion. Examples are given to illustrate these ideas. Throughout this paper p and v will be finite Borel measures on Hausdorff spaces A and Y, respectively. That is, p and v will be countably additive, nonnegative real-valued measures on "35(A) and %(Y), where ÇÔ(A) and %(Y) are the Borel sets of A and Y, respectively. Sometimes p will be regular, which is to say that it is inner regular with respect to the compact sets. At times v will be two-valued, which means that its range consists of the two values 0 and 1. If »»can be expressed as the sum of a countable collection of multiples of two-valued Borel measures, then v is purely atomic. A (not necessarily regular) Borel measure p will be called r-additive if p(C\ Fa) = inf p(Fa) whenever {Fa} is a decreasing family of closed sets [4, p. 96]. Equiva- lently, p is r-additive if p( U Va) = sup p( Va) for each increasingly directed family { Va} of open sets. Of course, each regular Borel measure is T-additive. A Borel rectangle is a set A X B such that A and B are Borel sets, and <3b(X) X ®(y) denotes the a-algebra generated by Borel rectangles. The notation p X v is reserved for the usual (not necessarily complete) product of p and v, with <$>(X)X %(Y) for its domain. If M is a Borel set in A X Y, let Mx = {y: (x,y) G M} for each x G X and let My = {x: (x,y) G M} for each y E Y. If M G % (X X y) and v(Mx) is measura- ble as a function in x (or p-almost everywhere equal to a measurable function), then we say that p X, v(M) is defined and we let p X, v(M) = f v(Mx) dp. If v is r-additive or if A is first countable, then Lemma 1.1 shows that v(Wx) is lower semicontinuous for each open set W in X X Y. Hence, if v is regular or if A is first countable, then v(Mx) is measurable as a function in x for all M G ® (A X Y) so that p x, v is a Borel measure on A X Y. In the same manner, if Received by the editors January 27, 1981. 1980 Mathematics Subject Classification. Primary 28A35; Secondary 28C15. Key words and phrases. Borel measure, purely atomic measure, separable-regular Borel measure, completion of a measure. © 1982 American Mathematical Society 0002-9947/82/0000-0777/$04.75 611 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 612 R. A. JOHNSON M G%(X X Y) and fi(My) is measurable (or »«-almost everywhere equal to a measurable function), then p x 2 v(M) is defined and fi X2v(M) = f p(My)dv. Notice that p X, v(M) and p X2 v(M) are defined and equal to p X v(M) for all M £ %(X)X <$>(Y). The main purpose of this paper is to give conditions in §3 such that each Borel set of A X y can be sandwiched between two members of © (A) X © ( Y) with the same p X »»-measure. (This happens, for example, if p is inner regular with respect to the separable compact sets, if A is first countable, and if v is purely atomic.) In that case, p X, v(M) and p X2 v(M) are defined and equal for all Borel sets M in A X y. The existence and equality of p X, v(M) and p x2 v(M) for all M G © (A X Y) are of interest because they make possible the existence and equality of both iterated integrals for each nonnegative Borel function on A X y [7, summary, p. 124]. I wish to thank D. H. Fremlin for helpful discussions concerning the ideas of this paper. In particular, Theorem 2.4 is due to him. 1. The p x j v-measure of an open set. In this section conditions are given under which each open set in A X y contains a set in ©(A) X %(Y) with the same p X x f-measure. For example, Theorem 1.2 shows that such containment occurs if A is first countable and p is regular. Lemma 1.1. Suppose v is a Borel measure on Y, and suppose W is an open set in X X Y. Let p G X, and let {Ua: a E A} be a base of open neighborhoods of p. If v( Wp) > k, then there exist a G A and an open set V such that Uax V G W and v( V) > k if either of the following conditions holds : (1) If k = card(/l), then the union of k Borel sets of v-measure zero has v-outer measure zero. (Of course, this condition is satisfied if A is countable.) (2) v(\J Vß) = sup v(Vß)for each increasingly directed family {Vß} of open sets in Y. That is, v is r-additive. Proof. Suppose W is open in A X y and that v( Wp) > k. For each y G Wp, choose an open neighborhood V(y) of y such that Ua X V(y) G W for some a G A. For each a G A, let Va be the union of those V(y) such that Ua X V(y) G W. Then Vp G Vy whenever Uß D Uy, and Wp = (J {Va: a G A}. Hence, there is a G A such that v( Va) > k. Since Ua X Va G W, we are done. Theorem 1.2. Suppose p is a regular Borel measure on X and that v is a Borel measure on Y. Then each open set in X X Y contains an open set in © (A) X % ( Y) with the same p X, v-measure if any one of the following conditions holds: (1) A is first countable. (2) The character (minimum cardinality of a neighborhood base) of each point in X is less than k, and the union of fewer than k Borel sets of v-measure zero has v-outer measure zero. (3) v is r-additive. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use PRODUCTS OF TWO BOREL MEASURES 613 Proof. Let W be open in A X Y, and fix n. By Lusin's theorem, there exists a compact set A in A such that the restriction of the function v(Wx) to A is continuous and such that p(X \ K) < l/n [6, Exercise 55.3]. For eachp in A, we may choose by Lemma 1.1 open sets U(p) and V(p) such that (i)p E U(p), (ii) v(Wx) <v(Wp)+ l/n for all x G U(p) n A, (iii)v(V(p))>v(Wp)- l/«,and (iv) U(p) X V(p) G W. Since K is compact, there exists a finite collection of points {p,} in A such that A is covered by the corresponding collection of open sets {U(p¡)}. If W(n) = \J{U(p,)X V(Pi)}, then W(n) is an open subset of Wsuch that W(ri) G%(X) X <$>(Y)and such that p X, v(W)<p Xxv(W(n)) + (2/n)p(X) + (l/n)v(Y). Then W* = U W(ri) is the required open subset of W with the same p X, v- measure as W. The conclusion of Lemma 1.1 clearly holds if v is regular. As a matter of fact, almost the same lemma can be proved if open sets of Y are inner regular with respect to a class of sets having a certain type of property provided that the only sets in A with that property are the closed sets. In order to make these ideas precise, let us say that a topological property P is compact-like if whenever a set A has property P, then so does (i) the product of any compact set with A, (ii) each closed subset of A, and (hi) the image of A under every continuous, open mapping. For example, countable compactness is a compact-like property. Now a set A is called N0- bounded (or strongly countably compact) if the closure in A of each of its countable subsets is compact ([5, p. 201] or [12, pp. 763-764]). The property of Nn- boundedness is also a compact-like property. Lemma 1.3. Suppose v is a Borel measure on Y such that each open set in Y contains a set with a given compact-like property P and with arbitrarily close v-outer measure. Suppose that every set in X with property P is closed.
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