Abstract: This paper discussed the construction of metric space on the nested geometrical object. Given a nested geometrical object K in Rⁿ and a continuous gauge function h(t), a new metric p was constructed on K such that 0<H^h (K)<+∞ in the new metric space (K, p). Particularly, if the gauge function is h(t)=t^s, then for any positive finite number s, it's also possible to construct a new metric p on K such that ~$\mathscrH^h (K)=1 and
dim_PK=dim_BK=dim_HK=s.

Key words: Hausdorff measure, metric, nested geometrical object, Hausdorff measure, metric