Is this c the same as that c?

Question

Are the highlighted $c$'s the same or should it be $c_1$ and $c_2$.

Answer 1

I am not very good at analysis, but the answer to your question is clearly "the $c$'s are the same", by plain elementary logic. I'll try to explain it to you: call $\varphi(\alpha)$ the "sentence" $$\forall x \ \left[ \overline{\lim}_{r \to 0} \mu(B(x,r)/r^s < \alpha \right]$$ and call $\psi(\beta)$ the "sentence" $$\mathcal H^s(F) \geq \mu(F)/\beta.$$ Then your theorem can be written as $$\varphi(c) \implies \psi(c).$$ If you think about it, you can see that it would make no sense to write something like $$\varphi(c_1) \implies \psi(c_2).$$

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I'll use an easier example for you to understand better: suppose $\alpha,\beta \in \mathbb N$ and call $\varphi(\alpha)$ the "sentence" $$x \leq \alpha.$$ Now call $\psi(\beta)$ the "sentence" $$x \leq 2\beta.$$ Then it is of course true that $$\varphi(n) \implies \psi(n),$$ since this can be written in natural language as

If $x \leq n$, then $x \leq 2n$.

But which sense would it make to say $\varphi(n_1) \implies \psi(n_2)$? That is, which sense would it make to state

If $x \leq n_1$, then $x \leq 2n_2$.

?