High Quality Content by WIKIPEDIA articles! In measure theory, tangent measures are used to study the local behavior of Radon measures, in much the same way as tangent spaces are used to study the local behavior of differentiable manifolds. Tangent measures are a useful tool in geometric measure theory. For example, they are used in proving Marstrand's theorem. Consider a Radon measure ? defined on an open subset ? of n-dimensional Euclidean space Rn and let a be an arbitrary point in ?. We can “zoom in” on a small open ball of radius r around a, Br(a), via the transformation T_{a,r}(x)=frac{x-a}{r}, which enlarges the ball of radius r about a to a ball of radius 1 centered at 0. With this, we may now zoom in on how ? behaves on Br(a) by looking at the push-forward measure defined by T_{a,r #}mu(A)=mu(a+rA) where a+rA={a+rx:xin A}.

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