### Notations:

- – Measurable space, (, ) where denotes the -algebra
- We mean measure, to be outer measure, that is function on to such that it is monotone, countably sub-additive and assumes value zero for the empty set.
- A measure is called Borel measure if the Borel sets (sets formed by open/closed sets) are measurable, that is in .
- A measure is Borel if and only if for any sets with we have . ( denotes the distance between two sets and )

- A Borel measure is Borel regular if for every there exists a Borel set such that and .
- If a Borel measure is not Borel regular, then defined as Borel , is Borel regular.

## Equivalent definitions of Hausdorff dimension

### 1. Definition of Hausdorff dimension:

(By Hausdorff via Caratheodory’s construction)

For any set we define the -dimensional Hausdorff measure of as,

where denotes the infimum of taken over all the collection of Borel cover of with diameter of , .

- Note that for all .
- When is an integer, denotes the volume of the -dimensional unit ball in (for ).
- For convenience, we assume to be one when we consider to be non-integer.
- is counting measure.

We observe the following:

- is Borel measure
- It is easy to check that is monotone, countably sub-additive non-negative measure that assumes value zero for the empty set, that is is a (outer) measure for all . Hence for any ,
- If with , then choose and by the definition of infimum choose a Borel cover of with , that is diameter of is . Then can either intersect or , but not both. So we have

and thus .

- That is, and hence . Thus is Borel measure.

- is Borel regular.
- Fix . Choose a Borel cover of such that . Then is a Borel set containing with .

- For and , implies . (In other words, implies )
- Note that for , by the definition of infimum, there exists a Borel cover of such that where denotes the diameter of . Hence

which goes to zero as goes to zero.

- Note that for , by the definition of infimum, there exists a Borel cover of such that where denotes the diameter of . Hence

The * Hausdorff dimension* of a set is defined as

*We have computed the Hausdorff dimension of generalized Cantor set here.*

(to be updated) This blog is loosely based on the notes prepared while giving a lecture series during January 2017-April 2017 at IISER - Bhopal. The definitions, the notations, proofs, theorems are from the references: 1) W. F. Donoghue; Distributions and Fourier Transforms, Acad. Press, 1969. MR3363413 2) Pertti Mattila, Geometry of sets and measures in Euclidean spaces: Fractals and rectifiability, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. MR1333890 3) P. Mattila. Fourier Analysis and Hausdorff Dimension, Cambridge University Press, Cambridge, 2015. 4) Wolff T.H.: Lectures on Harmonic Analysis. University Lecture Series, 29. Amer. Math. Soc., Providence, RI (2003)