On the Hausdorff measure of Brownian paths in the plane
- 1 April 1961
- journal article
- Published by Cambridge University Press (CUP) in Mathematical Proceedings of the Cambridge Philosophical Society
- Vol. 57 (2) , 209-222
- https://doi.org/10.1017/s0305004100035076
Abstract
Ω will denote the space of all plane paths ω, so that ω is a short way of denoting the curve. we assume that there is a probability measure μ defined on a Borel fieldof (measurable) subsets of Ω, so that the system (Ω,, μ) forms a mathematical model for Brownian paths in the plane. [For details of the definition of μ, see for example (9).] letL(a,b; μ) be the plane set of pointsz(t, ω) fora≤t≤b. Then with probability 1,L(a,b; μ) is a continuous curve in the plane. The object of the present note is to consider the measure of this point setL(a,b; ω).Keywords
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