Well definedness of an arc length type functional

Setting

Let $$\Omega\subset\mathbb{R}^2$$ be open and bounded and for $$x,y\in\bar\Omega$$ denote by $$\begin{equation*} C^{x,y} := \{\sigma\in W^{1,\infty}([0,1], \bar\Omega): \sigma(0) = x, \sigma(1) = y\}\,. \end{equation*}$$ For $$\varphi$$ mapping $$\bar\Omega$$ to $$\mathbb{R}_+$$ define \begin{align*} L_\varphi(\sigma)&:= \int_0^1 \varphi(\sigma(t))|\dot\sigma(t)| \mathrm{d}t\\ c_\varphi(x,y) &:= \inf\{ L_\varphi(\sigma) \,|\, \sigma\in C^{x,y}\}\,. \end{align*}

Question

Which regularity assumptions on $$\varphi$$ suffice for $$L_\varphi$$ and $$c_\varphi$$ to be well defined? If $$\varphi$$ is less than lower semi-continuous, does the integral in the definition of $$L_\varphi$$ still exist?

Context

In this paper, the authors define $$L_\varphi$$ for continuous functions $$\varphi$$. After Proposition 3.1, they present a computation which uses $$\varphi\in L^p$$ and state that the computation is purely formal due to $$L_\varphi$$ and $$c_\varphi$$ not being well-defined for $$\varphi\in L^p$$. They then spend the elaborate paragraph §3.2 on extending both $$L_\varphi$$ and $$c_\varphi$$ to $$L^p$$. It's not clear to me why this is necessary.

I see the problem now: The set $$\sigma([0,1])$$ may be a null set and $$\varphi\in L^p$$ is only defined almost everywhere such that the integrand $$\varphi(\sigma(t))$$ is not well-defined in general.