## The space of functions of bounded variation on curves in metric measure spaces

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- by O. Martio PDF
- Conform. Geom. Dyn.
**20**(2016), 81-96 Request permission

## Abstract:

The approximation modulus, $AM$–modulus for short, was defined in an earlier paper by the author. In this paper it is shown that an $L^1(X)$–function $u$ in a metric measure space $(X,d,\nu )$ can be defined to be of bounded variation on $AM$–a.e. curve in $X$ without an approximation of $u$ by Lipschitz or Newtonian functions. The essential variation of $u$ on $AM$ almost every curve is bounded by a sequence of non-negative Borel functions in $L^1(X)$. The space of such functions is a Banach space and there is a Borel measure associated with $u$. For $X = \mathbf {R}^n$ this gives the classical space of BV functions.## References

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## Additional Information

**O. Martio**- Affiliation: Department of Mathematics and Statistics, FI–00014 University of Helsinki, Finland
- MR Author ID: 120710
- Email: olli.martio@helsinki.fi
- Received by editor(s): September 24, 2015
- Received by editor(s) in revised form: February 10, 2016
- Published electronically: April 19, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Conform. Geom. Dyn.
**20**(2016), 81-96 - MSC (2010): Primary 26A45; Secondary 26B30, 30L99
- DOI: https://doi.org/10.1090/ecgd/291
- MathSciNet review: 3488026

Dedicated: Dedicated to Jan Malý on his $60^{th}$ birthday