In this paper, weak convergences of marked empirical processes in $L^2(\mathbb{R},ν)$ and their applications to statistical goodness-of-fit tests are provided, where $L^2(\mathbb{R},ν)$ is the set of equivalence classes of the square integrable functions on $\mathbb{R}$ with respect to a finite Borel measure $ν$. The results obtained in our framework of weak convergences are, in the topological sense, weaker than those in the Skorokhod topology on a space of cádlág functions or the uniform topology on a space of bounded functions, which have been well studied in previous works. However, our results have the following merits: (1) avoiding conditions which do not suit for our purpose; (2) treating a weight function which makes us possible to propose an Anderson--Darling type test statistics for goodness-of-fit tests. Indeed, the applications presented in this paper are novel.