Real Analysis Exchange

Convergence of Metric Space-Valued BV Functions

Isidore Fleischer and John E. Porter

Full-text: Open access


Chistyakov has proved ``Helly's selection theorem'' - a uniformly BV sequence has a pointwise convergent subsequence - for Banach-(resp. continuous, group-) valued functions from a real interval into a compact subset. We extend, dispensing with continuity, to arbitrary real subsets and lighten compactness of the range to pointwise precompactness (which answers one of his questions). In addition, we accomplish his selection more generally for complete metric-set-valued BV maps with closed graphs which are pointwise compact on dense subsets of their domains.

Article information

Real Anal. Exchange, Volume 27, Number 1 (2001), 315-320.

First available in Project Euclid: 6 June 2008

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A45: Functions of bounded variation, generalizations 54C65: Selections [See also 28B20] 54C60: Set-valued maps [See also 26E25, 28B20, 47H04, 58C06]

bounded variation pointwise precompact


Fleischer, Isidore; Porter, John E. Convergence of Metric Space-Valued BV Functions. Real Anal. Exchange 27 (2001), no. 1, 315--320.

Export citation


  • V. V. Chistyakov, On the theory of set–valued maps of bounded variation of one real variable, Matematicheskii Sbornik 189 (1998), No. 5, 153–176.
  • J. Dugundji, Topology, Allyn and Bacon, Inc., Boston, 1966.
  • H. Federer, Geometric Measure Theory, Springer–Verlag, New York, 1969.
  • J. J. Moreau, Bounded Variation in Time, in: J. J. Moreau, P. D. Panagiotopoulos, and G. Strang, eds., (Topics in Nonsmooth Mechanics, Birkhauser Verlag, 1988).
  • I. Natanson, Theory of Functions of a Real Variable, vol. I, New York (1974).