On the abscence of the Inner Measure in introductory texts on Measure Theory

Question

Letting $\mu$ and $\mu^*$ be the Lebesgue and outer Lebesgue measure repsectively, the inner Lebesgue measure can defined as $$\mu_*:S\mapsto \sup\left\{\mu(K) : \text{$K$ is a compact subset of $S$}\right\}$$ or, on bounded sets, as $$S\mapsto \mu(A)-\mu^*(A\setminus S)$$ for any elementary set $A$ containing $S$.

Neither Williams' Probabilities with Martingles, nor Hunter's Measure Theory, nor Tao's An Introduction to Measure Theory define inner measures in general. Why are inner measures skipped when studying introductory measure theory?

A definition of an inner measure may be found in Wikipedia, yet it's not clear at all why such a definition is adopted i.e. having the defined the Lebesgue inner measure, why are those properties in particular chosen when characterizing general inner measures? (see also the edit at the bottom of the post)

The outer measure is used in the construction of measures through -what I shall call- Carathéodory's Restriction Lemma (see below). Is there a version of this theorem which utilizes inner, as opposed to outer, measures?

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Carathéodory's Restriction Lemma: let $\mu^*:2^X\to [0,\infty]$ be an outer measure on the power set $2^X$ of $X$. Then $\mu^*$ can be restricted to a measure $$\mu:\Sigma\to[0,\infty]$$ where $\Sigma := \Big\{ C \in 2^X : C \text{ is Caratheodory measurable} \Big\}$ and $\mu := \mu^*|_{\Sigma}$.

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Edit: in Halmos' Measure Theory (where I was directed to by Dave L. Renfro below), an inner measure $\mu_*$ induced by a measure $\mu:\Sigma\to[0,\infty]$ is defined by \begin{equation} \tag{1} \mu_*(E):=\sup\left\{\mu(F):E\supseteq F \text{ for } F\in \Sigma\right\} \end{equation} and is proven to have the following two properties: \begin{equation} \begin{split} & a) \ \mu_*(\varnothing)=0.\\ & b) \ \mu_*(E)\le \mu_*(F) \text{ whenever } E\subseteq F.\\ & c) \ \text{For a disjoint sequence of sets } E_1, E_2, \ldots \text{ we have}\\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mu_*\left(\bigcup_{n=1}^{\infty}E_n\right) \ge \sum_{n=1}^{\infty}\mu_*(E_n) \end{split} \end{equation} (all sets in questions are subsets of $X$, where $\Sigma$ is a $\sigma$-algebra on $X$). Both $(1)$ or the combination of $a), b)$, and $c)$ are symmetrical to a characterization of the outer measure, which adds more weight to the question of why inner measures are defined instead so assymetrically (relative to outer measures) in Wikipedia.