Suppose I am given some continuous $1$- form $\omega$ defined on $\mathbb{R}^2$, and a fixed curve $C$ that is the boundary of a bounded open set $\Omega$. Let $\{C_j\}^\infty_{j=1}$ be some sequence of $C^1$ curves.
In what "sense" of convergence of $C_j$ to $C$ would imply that $\int_{C_j}\omega$ converges to $\int_C \omega$ ?
Given $C$ and $\Omega$ could I always find $C_j$ contained in $\Omega$ that converge to $C$ in this "sense"?