# real analysis – Doubt in Proposition \$3.3(a)\$ in Falconer’s Fractal Geometry (A Property of Hausdorff Dimension)

In what follows, $$mathcal H^s(F)$$ denotes the $$s$$-dimensional Hausdorff measure of $$Fsubsetmathbb R^n$$, and $$dim_H F$$ denotes the Hausdorff dimension of $$Fsubsetmathbb R^n$$.

Proposition $$3.3(a)$$:
Let $$Fsubsetmathbb R^n$$ and suppose that $$f ∶ F → ℝ^m$$ satisfies the Hölder condition $$|f (x) − f (y)| ⩽ c|x − y|^𝛼 quad (x,yin F)$$
Then $$dim_H f(F) ⩽ (1∕𝛼)dim_H F$$. In particular, if $$f$$ is a Lipschitz mapping, that is, if $$𝛼 = 1$$, then $$dim_H f (F) ⩽ dim_H F$$.

I think the proof given is incomplete, or I do not understand it well. Here it is:

If $$s > dim_H F$$, then by Proposition $$3.1$$, $$mathcal H^{s/𝛼}(f (F)) ⩽ c^{s∕𝛼}mathcal H^s(F) = 0$$,
implying that $$dim_H f (F) ⩽ s/𝛼$$ for all $$s > dim_H F$$. The conclusion for Lipschitz mappings is immediate on taking $$𝛼 = 1$$.

If $$s > dim_H F$$, I know that $$mathcal H^s(F) = 0$$. The conclusion from Proposition $$3.1$$ makes sense, and we get $$dim_H f (F) ⩽ s/𝛼$$ for all $$s > dim_H F$$. But, what about the cases $$s = dim_H F$$ and $$s < dim_H F$$? Do we not have to worry about those? Why? I’m not sure I understand how the author has proved the result, and I would appreciate any help.

For your reference, here is Proposition $$3.1$$:

Proposition $$3.1$$:
Let $$F ⊂ ℝ^n$$ and $$f ∶ F → ℝ^m$$ be a mapping such that $$|f (x) − f (y)| ⩽ c|x − y|^𝛼quad (x,yin F)$$
for constants $$𝛼 > 0$$ and $$c > 0$$. Then for each $$s$$, $$mathcal H^{s/𝛼}(f(F)) ⩽ c^{s/𝛼}mathcal H^s(F)$$
In particular, if $$f$$ is a Lipschitz mapping, that is, if $$𝛼 = 1$$, then $$mathcal H^s(f (F)) ⩽ c^smathcal H^s(F)$$