By Tom Lindstrom

ISBN-10: 0821824848

ISBN-13: 9780821824849

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**Example text**

57 BROWNIAN MOTION ON NESTED FRACTALS Nonstandard measure theory provides a straight-forward proof of the following crucial lemma. For the necessary background information, see Chapter 3 of [l ]. 4 Lemma. (T ,d ) is complete. Proof: If {x } ^ is a Cauchy-sequence, let n n£ tsf nonstandard version. Each ~k a d d i t i v e measure on A T(A)=L(TN) st x n is an internal, *-countably 2 ~k R . Choose an i n f i n i t e e l e m e n t N£ IN, + A be the Loeb-measure of xXT, and define a measure x N let L(x^T) N on R by where — A {x J1 * fcT be its n n ^ IN (st (A)), is the standard part map.

1 belongs to A the n-complex C associated with an n-cell C. 6C- s is the C, then BROWNIAN MOTION ON NESTED FRACTALS 37 A Proof: Assume that with an n-cell another, s belongs to the n-complex D associated D. Since the walk moves from one N-neighbor to A s, „ must also be an element of k-1 D. But then sk_1€COD = COD by the Nesting Axiom. The last topic I shall address in this section is the question of how many n-cells an element in E may belong to. 11. 13 Proposition. , , where i i 4> €¥ i has belongs to another n-cell and F.

V y But then . , F. x V^'V as a fixed point. If . , then " ',3n x x=cp. "" for ^ ' ' ##:3n belongs to four 2n-cells and also . (y) 3 also belongs to two n-cells x . C; just let F. D F K, • • , K F. , 11 • •# l i V •*' n, T "'n . Repeating the argument, we can make x an element of as many N-cells as we wish by just choosing N large enough. 3. 1 that denotes topological closure). For each closed ball with center z ECV (as usual, the bar z€F, let and radius 1. Since B z z€V, be the the 38 TOM LINDSTR0M intersection B flV must h a v e p o s i t i v e Vol (B flV) z V o l (B ) volume, and hence > j^ K z for some positive integer belongs to and let If B z.

### Brownian Motion on Nested Fractals by Tom Lindstrom

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