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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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: Suppose f is a continuous mapping with finite Lebesgue area from a polyhedral region $ X \subset {R^k}$ into $ {R^n},2 \leqq k \leqq n$. Let $ f = l \circ m$ 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 $ \{ {f_i}\} $ is any sequence of quasi-linear maps $ {f_i}:X \to {R^n}$ converging uniformly to f with bounded areas, then

$\displaystyle T(g)(\phi ) = \mathop {\lim }\limits_{i \to \infty } \int_X {(g \circ m)f_i^\char93 \phi } $

whenever $ \phi $ is an infinitely differentiable k-form in $ {R^n}$ and g is a continuous real valued function on M which vanishes on $ m ($Bdry$ \; X)$.

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

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Article copyright: © Copyright 1972 American Mathematical Society