## Real Analysis Exchange

### Conditions for equality of Hausdorff and packing measures on ℝ

H. Joyce

#### Abstract

This note answers the question, for which Hausdorff functions $h$ may the $h$-Hausdorff and $h$-packing measures agree on some subset $A$ of $\mathbb{R}^n$, and be positive and finite. We show that these conditions imply that $h$ is a regular density function, in the sense of Preiss, and that for each such function there is a subset of $\mathbb{R}^n$ on which the $h$-Hausdorff and $h$-packing measures agree and are positive and finite.

#### Article information

Source
Real Anal. Exchange, Volume 22, Number 1 (1996), 142-152.

Dates
First available in Project Euclid: 1 June 2012

https://projecteuclid.org/euclid.rae/1338515209

Mathematical Reviews number (MathSciNet)
MR1433602

Zentralblatt MATH identifier
0879.28014

Keywords
Measure and Integration

#### Citation

Joyce, H. Conditions for equality of Hausdorff and packing measures on ℝ. Real Anal. Exchange 22 (1996), no. 1, 142--152. https://projecteuclid.org/euclid.rae/1338515209

#### References

• K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications. John Wiley, 1990.
• H. Federer, Geometric Measure Theory. Springer-Verlag, 1969.
• P. Mattila, Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, 1995.
• P. Mattila and D. Preiss, Rectifiable measures in $\reals ^n$ and existence of principal values for singular integrals. J. London Math. Soc. (2) 52 \nolinebreak(1995), 482–496.
• A. P. Morse, Perfect Blankets. Trans. Amer. Math. Soc. 61 \nolinebreak(1947), 418–442.
• D. Preiss, Geometry of measures in $\reals ^n$: Distributions, rectifiability and densities. Ann. Math. 125 \nolinebreak(1987), 537–643.
• X. Saint Raymond and C. Tricot, Packing regularity of sets in $n$-space. Math. Proc. Camb. Phil. Soc. 103 \nolinebreak(1988), 133–145.
• S. J. Taylor and C. Tricot, Packing measure and its evaluation for Brownian paths. Trans. Amer. Math. Soc. 288 \nolinebreak(1985), 679–699.
• S. J. Taylor and C. Tricot, The packing measure of rectifiable subsets of the plane. Math. Proc. Camb. Phil. Soc. 99 \nolinebreak(1986), 285–296.
• C. Tricot, Two definitions of fractional dimension. Math. Proc. Camb. Phil. Soc. 91 \nolinebreak(1982), 57–74.