부산대학교 도서관

This important paper is concerned with two independent subjects mentioned in its title. More precisely, the first is a well-known problem of D. Maharam (1947); the other is a recent problem due to Kalton [Measure theory and its applications (De Kalb, Ill., 1980), see p. 284, Problem 5, Northern Illinois Univ., De Kalb, Ill., 1981; see MR 82g:28002]. Both problems are of importance for the theory of $F$-spaces (i.e., complete metrizable topological vector spaces). \par Let ${\cal A}$ be an algebra of sets. A set function $\varphi\colon{\cal A}\to{\bf R}$ is a submeasure [measure] if (i) $\varphi(\varnothing)=0$, (ii) $\varphi(A)\leq \varphi(B)$ whenever $A,B\in{\cal A}$ and $A\subset B$, (iii) $\varphi(A\cup B)\leq\varphi(A)+\varphi(B)$ $[\varphi(A\cup B)=\varphi(A)+\varphi(B)]$ whenever $A,B\in{\cal A}$ are disjoint. The submeasure $\varphi$ is said to be exhaustive [uniformly exhaustive] if $\lim\sb {n\to\infty}\varphi(A\sb n)=0$ for every disjoint sequence $(A\sb n\colon n\in{\bf N})$ in ${\cal A}$ [given $\varepsilon>0$ there exists $n\in{\bf N}$ such that $\min\sb {1\leq i\leq n}\varphi(A\sb i)\leq\varepsilon$ for any disjoint sets $A\sb 1,\cdots,A\sb n\in{\cal A}$]. \{The notion of uniform exhaustivity is due, in a related context, to M. Takahashi [Proc. Japan Acad. 42 (1966), 330--334; MR 34\#1476].\}Two submeasures $\varphi$ and $\psi$ on ${\cal A}$ are equivalent provided $\varphi(A\sb n)\to 0$ if and only if $\psi(A\sb n)\to 0$ for every sequence $(A\sb n\colon n\in{\bf N})$ in ${\cal A}$. Maharam's problem asks, in one of its equivalent forms, whether every exhaustive submeasure is equivalent to a measure. \{See papers by M. Sh. Goldshtein [Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk 1976, no. 5, 19--23; MR 55\#8307] and V. N. Aleksyuk [Mat. Zametki 21 (1977), no. 5, 597--604; MR 57\#16535] for some other equivalent forms.\}The authors prove that the answer is positive for a uniformly exhaustive submeasure (Theorem 3.4). This solves a folklore problem which was raised in a special case by M. Talagrand [Math. Ann. 252 (1979/80), no. 2, 97--102; MR 81k:28005]. Thus Maharam's problem is reduced to the following one: Does there exist an exhaustive submeasure which is not uniformly exhaustive? \{The latter problem seems to have been first published by I. Dobrakov [Dissertationes Math. (Rozprawy Mat.) 112 (1974), see p. 27; MR 51\#3382].\} As an application of Theorem 3.4, it is shown that a countably additive measure $\mu$ on a $\sigma$-algebra of sets with values in an $F$-space has a control measure provided the range of $\mu$ is relatively compact. \par A function $f\colon{\cal A}\to{\bf R}$ is said to be $\Delta$-approximately additive if $f(\varnothing)=0$ and given $A,B\in{\cal A}$ disjoint we have $\vert f(A\cup B)-f(A)-f(B)\vert \leq\Delta$. In answer to the problem of Kalton, it is proved that there exists a universal constant $K<45$ with the property that if $f\colon{\cal A}\to{\bf R}$ is $\Delta$-approximately additive, then there is an additive function $\mu\colon {\cal A}\to{\bf R}$ with $\vert f(A)-\mu(A)\vert \leq K\Delta$ (Theorem 4.1). This implies that $c\sb 0$ and, more generally, every ${\cal L}\sb \infty$-space is a ${\cal K}$-space. (An $F$-space $X$ is said to be a ${\cal K}$-space if whenever $Y$ is an $F$-space with a one-dimensional subspace $L\subset Y$ such that $Y/L\cong X$, then $L$ is complemented in $Y$; see a paper by Kalton [Compositio Math. 37 (1978), no. 3, 243--276; MR 80j:46005].) \par The proofs of Theorems 3.4 and 4.1 are based on the existence of an $(m,p,q,r)$-concentrator $R$, where $m\geq p\geq q\geq r$ are suitably chosen natural numbers. The concentrator $R$ is a map which assigns to each natural number $j$, $1\leq j\leq m$, a subset of $\{1,\cdots,p\}$ so that (i) $\sum\sp m\sb {j=1}\vert R(j)\vert \leq rm$, (ii) $\vert \bigcup\sb {j\in E}R(j)\vert \geq\vert E\vert $ whenever $E\subset\{1,\cdots,m\}$ and $\vert E\vert \leq q$. The proof of Theorem 3.4 also applies to Hall's marriage lemma. \par \{Reviewer's remarks: (1) The first example of a pathological submeasure (i.e. a submeasure which does not dominate a nonzero measure) is due to V. A. Popov [Twenty-sixth Hercen lectures: mathematics (Russian), 98--102, Leningrad. Gos. Ped. Inst., Leningrad, 1973; RZhMat 1973:9 B 772]. (2) The notion of uniform exhaustivity was previously studied by Popov [Functions of sets (Russian), 40--49, Komi Gos. Ped. Inst., Syktyvkar, 1977; MR 82a:28007] and M. N. Lubyshev [ibid., 62--63; MR 81d:28003]. (3) Under a graph-theoretic hypothesis, the authors' Theorem 3.4 was obtained independently by Ch. Bandt [``Zum Zusammenhang von Mass und Metrik'', Dissertation, Ernst-Moritz-Arndt-Univ., Greifswald, 1982; per revr.]. (4) For a lower bound of the constant $K$ of Theorem 4.1 see B. Pawlik [``Approximately additive set functions'', Colloq. Math., to appear]. (5) The set function $A\mapsto f(A)$ in the proof of Proposition 6.2 is merely $2\Delta$-approximately additive, and so the constant ``100'' of that proposition should be increased. (6) Reference [9] by Kalton has appeared [Proc. Edinburgh Math. Soc. (2) 26 (1983), no. 1, 29--48; MR 84d:46046]. (7) For other recent results about pathological submeasures see another paper by the authors [Math. Ann. 262 (1983), no. 1, 125--132; MR 84d:28018].\}MR