The range of a vector measure

The purpose of this note is to prove that the range of a countably additive finite measure with values in a finite-dimensional real vector space is closed and, in the non-atomic case, convex. These results were first proved (in 1940) by A. Liapounoff. In 1945 K. R. Buch (independently) proved the part of the statement concerning closure for non-negative measures of dimension one or two. In 1947 I offered a proof of Buch's results which, however, was correct in the onedimensional case only. In this paper I present a simplified proof of LiapounofPs results. Let X be any set and let S be a <r-field of subsets of X (called the measurable set of X). A measure /i (or, more precisely, an iV-dimensional measure (#i, • • • , fix)) is a (bounded) countably additive function of the sets of S with values in iV-dimensional, real vector space (in which the "length" |&| + • • • + |£JV| of a vector ? = (?i» • • • » &v) is denoted by |£ | ) . The measure (/xi, • • • /*#) is non-negative if fXi(E) ^ 0 for every £ £ S and i«= 1, • • • , N. For a numerical (one-dimensional) measure /x0, Mo*0E) will denote the total variation of /xo on £ ; in general, if /x = (jiu • • • , Mtf), M* will denote the non-negative measure (jit!*, • • • , /$). The length |JU*| =Mi*+ • • • +M#* is always a non-negative numerical measure.