does the perimeter of a cell set containing a degenerate curve tend to zero, when the edge of the cells tends to zero?

[sergei akbarov]

Dear colleagues,

I wonder if anybody can clarify me the following. Suppose I is a
closed interval, f:I->R^2 an infinitely smooth map, and K a compact
set of zero measure in I. For any natural number k we take a net of
points in R^2 of the form (m/2^k,n/2^k), where m and n are arbitrary
integers, and every square in R^2 with vertices in such points and
with the edge 1/2^k we call a cell of order k. Then let B_k be a union
of cells of order k, which have a non-zero intersection with f(K). Is
is true that the perimeter of B_k tends to zero, as k tends to
infinity?

Thank you in advance,
Sergei Akbarov

[Dan Luecking]

I think so. The limit defines a measure (or at least an outer
measure) equivalent to the 1-dimensional Hausdorff measure
(that is, they have the same null sets). And the 1-dim Hausdorff
measure of f(K) is 0.

Dan
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