Let $\mathbf{S}$ be a collection of subsets of a set $\mathbf{X}$ and $\mu : \mathbf{S} \to [0,\infty]$ is a set function. Show that $\mu$ is countably monotone if and only if $\mu^*$ is an extension of $\mu$. (From Royden Fitpatrick Ch17.29)
For the reverse direction, assume $\mu^*$ is the extension of $\mu$. In other words, $\mu^*$ restricted to $\mathbf{S}$ equals $\mu$.
Since the outer measure $\mu^*$ satisfies $\mu^*$($\varnothing$) = $0$ and countable monotonicity, then $\mu$ being the restirction of $\mu^*$ to $\mathbf{S}$ also satisfies the above properties.
For the forward direction, I am not sure how showing a set function with countable monotonicity would imply the outer measure is an extension of itself.
Edit: The proof for the forward direction. Thanks Dr. Bob Gardnerr