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?



Density function approximation with respect to $L^1$ distance
From the [description](https://rads.stackoverflow.com/amzn/click/com/0471816469) of the book *Nonparametric Density Estimation: The L1 View* by Devroye and Gyorfi:



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> 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.
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