A Doubling Measure Can Charge a Rectifiable Curve
نویسندگان
چکیده
By reparametrization, one may assume that γ is Lipschitz with constant equal to length(γ). We will also make use of the following simple (and well-known) criterion: a compact set Γ is the image of a rectifiable curve if and only if it is connected and H(Γ) < ∞. Indeed, one may choose γ so that length(γ) ≤ CH(Γ); see, for example, [1, 2]. Here and below, H denotes the one-dimensional Hausdorff measure. The purpose of this note is to prove
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A Doubling Measure on R Can Charge a Rectifiable Curve
For d ≥ 2, we construct a doubling measure ν on Rd and a rectifiable curve Γ such that ν(Γ) > 0.
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