Decomposition of a compactum into small geometric measure sets (Q1199900)

The \(n\)-dimensional geometric measure \(\mu_ n(X)\) of a compactum \(X\) lying in \(\ell_ 2\), the Hilbert space of square summable sequences, was defined by \textit{K. Borsuk} [Ann. Pol. Math. 42, 17-24 (1983; Zbl 0556.28009)] to be the lower bound of all \(\alpha>0\) such that for each \(\varepsilon>0\) there is a continuous \(f:X\to\ell_ 2\) such that \(\text{dist}(x,f(x))<\varepsilon\) for all \(x\in X\) and \(f(X)\) is contained in a polyhedron \(P\) in \(\ell_ 2\) for which the elementary \(n\)-dimensional measure of \(P\) is less than or equal to \(\alpha\). In other work by \textit{K. Borsuk}, \textit{S. Nowak} and \textit{S. Spiez} [Fundam. Math. 121, 59-71 (1984; Zbl 0568.28008)] it was shown that the \(n\)- dimensional geometric measure is not sub-additive. In this paper it is shown that if \(\mu_ n(X)<\infty\) for a compactum \(X\subset\ell_ 2\), then for each \(\varepsilon>0\) there exist compacta \(A\) and \(B\) such that \(X=A\cup B\) and \(\mu_ n(A)+\mu_ n(B)<\varepsilon\). The method of proof is a direct construction of the decomposition.