Strictly Positive Hausdorff Measure implies Infinite Lower Dimensional Measure

Theorem

Let $n \in \N_{>0}$.

Let $F \subseteq \R^n$ be a subset of the real Euclidean space.

Let $\map {\HH^s} \cdot$ denote the $s$-dimensional Hausdorff measure.

Let $s \in \R_{\ge 0}$.

Then:

$\map {\HH^s} F > 0 \implies \forall t \in \hointr 0 s : \map {\HH^t} F = +\infty$

Proof

Let:

$\exists t \in \hointr 0 s : \map {\HH^t} F < +\infty$

Then by Finite Hausdorff Measure Implies Zero Higher Dimensional Measure:

$\map {\HH^s} F = 0$

Hence the result by Proof by Contraposition.

$\blacksquare$