Paths in ${\mathbb{R}}^n$, inscribed polygonal sums, arc length as their supremum, and rectifiability

Let $n\ge 1$ and $a\le b$. A path in ${\mathbb{R}}^n$ is a continuous map $\gamma :[a,b]\to {\mathbb{R}}^n$. The map, its domain, and its parametrization are part of the path; its trace is only the set $\gamma ([a,b])$.

If $a<b$ and $P=(t_0,\dots ,t_m)$ is a partition of $[a,b]$, define the polygonal length inscribed by $P$ by

${\mathrm{\ell }}_P(\gamma ):={\sum }_{i<m}\parallel \gamma (t_{i+1})-\gamma (t_i){\parallel }_2\mathrm{.}$

The arc length is the extended-real supremum

$L_{[a,b]}(\gamma ):={\sup }_P{\mathrm{\ell }}_P(\gamma )\in [0,+\mathrm{\infty }]\mathrm{.}$

The path is rectifiable when these polygonal lengths are bounded above in $\mathbb{R}$, equivalently when $L_{[a,b]}(\gamma )<+\mathrm{\infty }$. In that case the length is a nonnegative real number. When the interval is clear, write $L(\gamma )$.

On a singleton interval $[a,a]$, define $L_{[a,a]}(\gamma ):=0$ and call every path with that domain rectifiable. There is one such path for each point of ${\mathbb{R}}^n$, namely the map sending $a$ to that point. This convention does not invoke a partition, whose published definition assumes distinct endpoints.