Background
I have been looking at the line integral and I am wondering why (or even if) it is well-defined in the context of Lebesgue integrals.
The definition of wikipedia defines it like this (in terms of Riemann integrals, as far as I can tell):
For some scalar field $f:U \subseteq \mathbb{R}^n \to \mathbb{R}$, the line integral along a piecewise smooth curve $C \subseteq U$ is defined as $$\int_C f(\mathbf{r}(s)) ds = \int_a^b f(\mathbf{r}(t)) |\mathbf{r}'(t)| dt$$ where $\mathbf{r}:[a,b] \to C$ is an arbitrary bijective parametrization of the curve $C$ such that $\mathbf{r}(a)$ and $\mathbf{r}(b)$ give the endpoints of $C$ and $a<b$.
Note that the first integral in the definition is just notation, its meaning is defined in terms of an arbitrary parametrization. In order for the definition to be well-defined, it is important that any choice of parametrization gives the same result.
Question
If we are working with the Lebesgue integral, an appropriate definition of the line integral seems to be (feel free to suggest an alternative defintion)
For Lebesgue measurable $U \subseteq \mathbb{R}^n$ and Lebesgue measurable function $f:U \to \mathbb{R}$, the line integral along a piecewise smooth curve $C \subseteq U$ is defined as $$\int_C f(\mathbf{r}(s)) ds = \int_{[a,b]} f(\mathbf{r}(t)) |\mathbf{r}'(t)| dt$$ where $\mathbf{r}:[a,b] \to C$ is an arbitrary bijective parametrization of the curve $C$.
I relaxed the restriction on $f$ to be any Lebesgue measurable function, and removed the restriction that $\mathbf{r}(a)$ and $\mathbf{r}(b)$ must be endpoints of $C$, because the Lebesgue integral ignores the "order" in which we integrate over the space (we still integrate over the whole $C$ because $\mathbf{r}$ is bijective). Again, the first integral is just notation, and the definition of this notation is in terms of an arbitrary parametrization.
Is the definition in terms of the Lebesgue integral well-defined? Concretely, these questions come to mind:
- What should be the interpretation of smooth in these definitions? Suggestion based on the comment by @PaulSinclair: A set is a smooth curve iff it has a bijective parametrization that has continous first-order partial derivatives almost everywhere.
- Is the definition independent of the parametrization?
- Is the Lebesgue integral always defined, for any measurable function and any smooth curve?
If so, why (proof)? If not, what should I change such that the definition becomes well-defined? If there are any resources on this (books, ...), I am interested in that as well.