A natural generalization of Riemann-Stieltjes integration is Lebesgue integration. Some would say that the use of Lebesgue integration would be overkill when treating differentiable or continuous functions, theoretically, Lebesgue integration describes things much simpler than RS-integration.
Likewise, is there a natural generalization of line integrals?
The standard definition of line integration is as follows:
Let $U$ be open in $\mathbb{R}^n$.
Let $\gamma:[a,b]\rightarrow L(\subset U)$ be a path in $\mathbb{R}^n$.
Let $F:U\rightarrow \mathbb{R}^n$ be a function.
Then, $\int_\gamma f$ is defined as $\lim_{\Delta t\to 0} \sum_{i=1}^n \langle F(r(t_i)),\Delta r_i \rangle$ if it exsits.
I wonder if there exists a measure relates a path $\gamma$ to functions $f$, or other abstract definitions of line integrations.
Thank you in advance.