Non-regular tangential behaviour of a monotone measure

A Radon measure mu on R-n is said to be k-monotone if r bar right arrow mu(B(x,r))/r(k) is a non-decreasing function on (0, infinity) for every x is an element of R-n. (If mu is the k-dimensional Hausdorff measure restricted to a k-dimensional minimal surface then this important property is expressed by the monotonicity formula.) We give an example of a 1-monotone measure mu in R-2 with non-unique and non-conical tangent measures at a point. Furthermore, we show that mu can be the one-dimensional Hausdorff measure restricted to a closed set A subset of R-2.