Given iid samples $X_1,...,X_N$ drawn from some unknown distribution with not necessarily continuous density function $f(x)$ are there any theorems/papers where based on the data $X_1,...,X_N$ an estimator $f_N(x)$ of $f(x)$ is defined and the approximation rate $$\int_{\mathbb{R}}|f(x)-f_N(x)|dx$$ is estimated?

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From the description of the book *Nonparametric Density Estimation: The L1 View* by Devroye and Gyorfi:

The first systematic single-source examination of density estimates. It develops, from first principles, the ``natural'' theory for density estimation, L1, and shows why the classical L2 theory masks some fundamental properties of density estimates.