If $\mu : \mathscr{H} \to \mathbb{R}$ is a pre-measure on $\mathscr{H}\subset X$ a semiring and $\mu^*$ the outer measure defined by $\mu$, $\mathscr{A}=\sigma(\mathscr{H})$ and $\nu$ a measure such that $\nu|_\mathscr{H}=\mu$, show that $\nu(A)\leq\mu^*(A)$ for all $A\in\mathscr{A}$.
My attempt is to show that this holds for $A\in\mathscr{H}$, but obviously this does not generalize to all of $\mathscr{A}$.
Given any $A\in\mathscr{A}$, we have \begin{align*} \nu(A) &\leq \inf \left \{\sum_{n = 1}^\infty \nu(A_n): \forall n \in \Bbb N, \ A_n \in \mathscr{H}, \ A \subseteq \bigcup_{n = 1}^\infty A_n \right\} = \\ & = \inf \left \{\sum_{n = 1}^\infty \mu(A_n): \forall n \in \Bbb N, \ A_n \in \mathscr{H}, \ A \subseteq \bigcup_{n = 1}^\infty A_n \right\} = \\ &= \mu^*(A) \end{align*} So, for all $A\in\mathscr{A}$, $\nu(A) \leq \mu^*(A)$.