Functions of bounded variation and curves in metric measure spaces (Q323091)

Every BV-function \(u\) in \(\mathbb R^n\) (understood as representing a finite Radon measure) has the property that, for every coordinate, \(u\) is of essentially bounded variation along almost every line parallel to this coordinate axis; see \S\,5 of [\textit{L. C. Evans} and \textit{R. F. Gariepy}, Measure theory and fine properties of functions. Boca Raton: CRC Press (1992; Zbl 0804.28001)]. In the article under review, a similar (Fubini-type) results are considered in the setting of a metric measure space \(X\) instead of \(\mathbb R^n\).NEWLINENEWLINESince in \(X\) a coordinate structure is missing, as a gauge for negligibility, the author introduces the concept of \textit{AM-modulus} of a family \(\Gamma\) of (rectifiable) curves. A sequence of non-negative measurable functions \(u_n: X\to\overline{\mathbb R}\) is said to be AM-admissible for \(\Gamma\) if for every \(\gamma\in\Gamma\), NEWLINE\[NEWLINE \liminf_{n\to\infty}\int_\gamma u_n\,ds\geq1. NEWLINE\]NEWLINE Then, the AM-modulus of \(\Gamma\) is defined as NEWLINE\[NEWLINE \text{AM}(\Gamma)=\inf\left\{\liminf_{n\to\infty}\int_\gamma u_n\,d\nu\right\}, NEWLINE\]NEWLINE where \(\inf\) ranges over all sequences \((u_n)_{n=1}^\infty\) AM-admissible for \(\Gamma\). Introductory results concern basic properties of AM-modulus and its connection to the \textit{M\(_1\)-modulus}, a concept similar to the AM-modulus [\textit{B. Fuglede}, Acta Math. 98, 171--219 (1957; Zbl 0079.27703)]. Next, with the use of upper gradients, two semi-norms \(\|D_{\text{BV}}u\|(X)\) and \(\|D_{\text L}u\|(X)\) are introduced, whose finiteness can be recognized as bounded variation of \(u: X\to\overline{\mathbb R}\). Then, two main results of the paper follow. Provided \(u\) is finite almost everywhere and \(\|D_{\text{BV}}u\|(X)<\infty\), there is \(\Gamma\) with \(\text{AM}(\Gamma)=0\) such that for every rectifiable curve \(\gamma:[a,b]\to X\) not in \(\Gamma\), the function \(u\circ\gamma\) is of bounded variation (Theorem 4.1). If, on the other hand, \(u\) is summable and \(\|D_{\text L}u\|(X)<\infty\), then \(u\circ\gamma\) is of essentially bounded variation for all \(\gamma\notin\Gamma\), where, as above, \(\text{AM}(\Gamma)=0\) (Theorem 5.4).