PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

On the differentiation of integrals with respect to translation invariant convex density bases

Volume 246 / 2019

Giorgi Oniani Fundamenta Mathematicae 246 (2019), 205-216 MSC: 28A15, 42B25. DOI: 10.4064/fm613-8-2018 Published online: 18 February 2019

Abstract

For a translation invariant convex density basis $B$ it is shown that its Busemann–Feller extension $B_{\mathrm {BF}}$ has properties close to $B$, namely $B_{\mathrm {BF}}$ differentiates the same class of non-negative functions as $B$, and the integral of an arbitrary non-negative function $f\in L(\mathbb {R}^n)$ at almost every point $x\in \mathbb {R}^n$ has the same type limits of indeterminacy with respect to the bases $B$ and $B_{\mathrm {BF}}$. This theorem provides a certain general principle of extending results obtained for Busemann–Feller bases to results for bases without the Busemann–Feller property. Applications of the theorem are given.

Authors

  • Giorgi OnianiDepartment of Mathematics
    Akaki Tsereteli State University
    59 Tamar Mepe St.
    Kutaisi 4600, Georgia
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image