So the example he gave was using the ML-Inequality. Let $v = \{e^{i \theta} \mid 0 \leq \theta \leq 2\pi \}$.
$\left|\int(1/z)\,dz\,\right|$ with respect to $v$ is less than or equal to $\int \left|1/z\right| \left|dz\right|$ with respect to $v^*$. Then he says that is equal to the integral with respect to $v$ of $\left|dz\right|$ which is equal to the $l(v) = 2 \pi$. ( he says that $\left|1/z\right|$ is the unit circle. ) I don't understand why $\left|1/z\right|$ is the unit circle and any of the steps after he makes this claim.