Strictly Positive Hausdorff Measure implies Infinite Lower Dimensional Measure
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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.
$\blacksquare$