A~result on differentiable measures on a~linear space

A. V. Uglanov

Sbornik. Mathematics, Tome 29 (1976) no. 2, pp. 217-222

Résumé

The basic content of this note is the proof of the following result. Let $X$ be a linear space, let $L$ be a subspace of it with $\dim L=m\infty$, let $R$ be a ring of subsets of $X$ which is invariant with respect to shifts by vectors in $L$, and let $\sigma$ be a finitely additive bounded quasi-content on $R$ which is differentiable $n$ times with respect to the subspace $L$. Then, for any bounded set $W\subset L$, $$ \lim_{r\to0}\sup_{L^c}\frac{|\sigma|(rW+L^c)}{r^{mn/(m+n)}}=0, $$ where $L^c$ is a linear complement to $L$ with respect to $X$, and $|\sigma|$ is the total variation of the quasi-content $\sigma$. Bibliography: 2 titles.