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I'm reading book "Geometric function theory and non-linear analysis" and there is one thing that I did not understand.

A curve family $\Gamma$ is a collection of curves. An admissible density is a Borel function $\rho$ for which $$\int_\gamma\rho\,ds > \geqslant 1 \text{ for all } \gamma \in \Gamma$$

The modulus of $\Gamma$ is $$M(\Gamma) = > \operatorname{inf}\int_{\mathbb R^n}\rho^n(x)\,dx$$ where the infimum is over all admissible densities for $\Gamma$.

The property I do not understand is

If $\Gamma$ contains a single "constant curve", then $M(\Gamma) = > \infty$.

I thought "constant curve" is just a point in $\mathbb R^n$. But what are admissible densities in this way?

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up vote 1 down vote accepted

If $\gamma$ is a constant curve, the condition $\int_\gamma \rho\, ds\ge1$ cannot be satisfied because the integral is always zero. Therefore, there are no admissible densities for a curve family with a constant curve. In the definition of the modulus we have infimum taken over an empty set, and $\inf \emptyset =+\infty$.

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