Abstract:
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 $\sigma$.
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