What is the definition of $s$-dimensional Lebesgue measure on $R^d$ for $s\ne d$?

Question

On the Appendix C about Hausdorff dimensions and measures of the book Multiparamerer Processes by Khoshnevisan it is written the following

Lemma 1.1.2 For all Borel sets $E\subset R^d$, and for all real numbers $s\geq 0$, $$H_s(E)\geq Leb_s(E). $$

Proof Let us fix an arbitrary $\varepsilon>0$ and find arbitrarily closed $l^\infty$ balls $B_1, B_2...$ whose radii $r_1, r_2...$ are all less or equal to $\varepsilon$. Whenever $E\subset\cup_i B_i$ $$Leb_s(E)\leq\sum_i Leb_s(B_i)=\sum_i(2 r_i)^s $$ (...)

Where $H_s$ is of course the s-dimensional Hausdorff measure on $R^d$.

My question is the following: Do you know which is the definition of $Leb_s$? He refers to it as the $s$-dimensional Lebesgue measure on $R^d$.

I just know the standard definition of Lebesgue measure https://proofwiki.org/wiki/Definition:Lebesgue_Measure

I also checked on the rest of the book to see if there was a more concrete definition of $Leb_s$ and I didn´t find it... probably I missed it!

Answer 1

I finally contacted the author of the book and there is a typo. The correct statement is

Lemma 1.1.2 For all Borel sets $E\subset \mathbb{R}^d$, and for all real numbers $s\geq d$, $$H_s(E)\geq Leb_d(E). $$