By Camillo De Lellis

The characterization of rectifiable units during the lifestyles of densities is a pearl of geometric degree thought. The tough facts, as a result of Preiss, is dependent upon many appealing and deep rules and novel suggestions. a few of them have already confirmed invaluable in different contexts, while others haven't but been exploited. those notes provide an easy and brief presentation of the previous and supply a few viewpoint of the latter. this article emerged from a path on rectifiability given on the college of Zurich. it's addressed either to researchers and scholars; the one prerequisite is a high-quality wisdom in commonplace degree thought. the 1st 4 chapters supply an creation to rectifiable units and measures in Euclidean areas, protecting classical subject matters corresponding to the world formulation, the concept of Marstrand and the main hassle-free rectifiability criterions. The 5th bankruptcy is devoted to a sophisticated rectifiability criterion as a result of Marstrand and generalized via Mattila, and the final 3 concentrate on Preiss' end result. the purpose is to supply a self-contained reference for someone drawn to an outline of this interesting subject.

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**Additional info for Rectifiable Sets, Densities and Tangent Measures**

**Example text**

To prove the uniqueness of the decomposition, note that the intersection between a purely unrectifiable set and a rectifiable set has always H k measure 0. Therefore, if F r and F u are as in the statement of the theorem we have H k (E r ∩E u ) = H k (E r ∩F u ) = H k (F r ∩E u ) = H k (F r ∩F u ) = 0 . 3). We are now ready for the following two lemmas. The first is a trivial application of the Decomposition Theorem and of the Besicovitch Differentiation Theorem. 7) of the previous chapter. 5. 8.

Without loss of generality we can assume x = 0. Then there exist a positive η and a sequence r j ↓ 0 such that for every k-dimensional plane W and every j • either H k F ∩ Br j (0) \ z : dist (W, z) ≤ ηr j • or there exists z j,W ∈ W ∩ Br j (0) with H k F ∩ Bηr j (z j,W ) r kj ≤ ≥ ηr kj ; 1 . 34) ∗k Set ν j := r −k j ν0,r j . Since θ (ν, x) < ∞, we can assume that a subsequence (not relabeled) of {ν j } converges to ν ∞ ∈ Tank (ν, x). 32) it follows that for some k-dimensional linear plane W and some constant c¯ ≥ 0 we have ν ∞ = c¯H k W .

Let us begin by assuming the contrary. Then there would exist z ∈ C ∩ B r (a) with dist (z, a + W ) > ηr . Therefore, Bηr/2 (z) would be contained in B 2r (a) \ z : dist (z, a + W ) ≤ ηr/2 . 10). H k E ∩ B2r (a) \ z : dist (z, a + W ) ≤ ηr/2 < δ Step 2 Now let us fix an arbitrary k-dimensional linear plane V . For each i ∈ N define the sets Ci := a ∈ C : C ∩ Bi −1 (a) \ C(a, V , η−1) = ∅ . 7, the intersection of C i with a ball of radius i −1 /2 is contained in a Lipschitz graph. Therefore, since C is purely unrectifiable, we have Hk Ci = 0.