J. London Math. Soc. (2) 80 (2009) 514–530 C2009 London Mathematical Society doi:10.1112/jlms/jdp040 Packing dimension of mean porous measures

We prove that the packing dimension of any mean porous Radon measure on Rd may be estimated from above by a function which depends on mean porosity. The upper bound tends to d − 1 as mean porosity tends to its maximum value. This result was stated in D. B. Beliaev and S. K. Smirnov [‘On dimension of porous measures’, Math. Ann. 323 (2002) 123–141], and in a weaker form in E. Järvenpäa ̈ and M. Järvenpäa ̈ [‘Porous measures on Rn: local structure and dimensional properties’, Proc. Amer. Math. Soc. (2) 130 (2002) 419–426], but the proofs are not correct. Quite surprisingly, it turns out that mean porous measures are not necessarily approximable by mean porous sets. We verify this by constructing an example of a mean porous measure μ on R such that μ(A) = 0 for all mean porous sets A ⊂ R. 1