Geöcze area and a convergence property

Author:
Ronald Gariepy

Journal:
Proc. Amer. Math. Soc. **34** (1972), 469-474

MSC:
Primary 28A75; Secondary 26A63

MathSciNet review:
0297974

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Abstract | References | Similar Articles | Additional Information

Abstract: Suppose *f* is a continuous mapping with finite Lebesgue area from a polyhedral region into . Let be the monotone-light factorization of *f* with middle space *M*.

If *f* satisfies a ``cylindrical condition'' considered by T. Nishiura, then a current valued measure *T* over *M* can be associated with *f* by means of the Cesari-Weierstrass integral, and if is any sequence of quasi-linear maps converging uniformly to *f* with bounded areas, then

*k*-form in and

*g*is a continuous real valued function on

*M*which vanishes on Bdry.

The total variation measure of *T*, taken with respect to mass, coincides with the Geöcze area measure over *M*.

**[C1]**Lamberto Cesari,*Quasi-additive set functions and the concept of integral over a variety*, Trans. Amer. Math. Soc.**102**(1962), 94–113. MR**0142723**, 10.1090/S0002-9947-1962-0142723-9**[C2]**Lamberto Cesari,*Extension problem for quasi-additive set functions and Radon-Nikodym derivatives*, Trans. Amer. Math. Soc.**102**(1962), 114–146. MR**0142724**, 10.1090/S0002-9947-1962-0142724-0**[F1]**Hebert Federer,*Currents and area*, Trans. Amer. Math. Soc.**98**(1961), 204–233. MR**0123682**, 10.1090/S0002-9947-1961-0123682-0**[F2]**Herbert Federer,*Some theorems on integral currents*, Trans. Amer. Math. Soc.**117**(1965), 43–67. MR**0168727**, 10.1090/S0002-9947-1965-0168727-0**[G]**Ronald Gariepy,*Current valued measures and Geöcze area*, Trans. Amer. Math. Soc.**166**(1972), 133–146. MR**0293066**, 10.1090/S0002-9947-1972-0293066-0**[N1]**Togo Nishiura,*The Geöcze 𝑘-area and a cylindrical property*, Proc. Amer. Math. Soc.**12**(1961), 795–800. MR**0125941**, 10.1090/S0002-9939-1961-0125941-X**[N2]**Togo Nishiura,*Integrals over a product variety and Fubini theorems*, Rend. Circ. Mat. Palermo (2)**14**(1965), 207–236. MR**0197685****[N3]**Togo Nishiura,*Area measure and Radó’s lower area*, Trans. Amer. Math. Soc.**159**(1971), 355–367. MR**0281880**, 10.1090/S0002-9947-1971-0281880-6

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1972-0297974-1

Article copyright:
© Copyright 1972
American Mathematical Society