I'm an engineer who self-studies pure math in my free time, so I apologise in advance if this is a silly question. I am currently reading "Analysis on Manifolds" by James Munkres. Theorem 14.1 states that "A subset $S$ of $\mathbb{R}^n$ is rectifiable if and only if $S$ is bounded and Bd$S$ has measure zero". Here, Bd$S$ refers to the boundary of $S$, and is defined on p.29 as follows:
"Let $S$ be a subset of $\mathbb R^n$. The interior of $S$, as a subset of $\mathbb R^n$, is defined to be the union of all open sets of $\mathbb R^n$ that are contained in $S$; it is denoted Int$S$. The exterior of $S$ is defined to be the union of all open sets of $\mathbb R^n$ that are disjoint from $S$; it is denoted Ext$S$. The boundary of $S$ consists of those points of $\mathbb R^n$ that belong neither to Int$S$ nor to Ext$S$ ; it is denoted Bd$S$."
Theorem 14.1 seems to me to imply that there can be subsets $S$ of $\mathbb{R}^n$ where Bd$S$ does not have measure zero in $\mathbb{R}^n$. This is what is confusing me. My intuition suggests that for any $S\subset\mathbb{R}^n$, Bd$S$ will be an $(n-1)$-manifold whose hyper-volume is zero in $\mathbb{R}^n$. Therefore, Bd$S$ will always have measure zero. Can anyone help me resolve this confusion? Do I need to have studied measure theory to fully understand this?