Let $f\in L^p(\mathbb{R})$ be a non-negative function. Being in $L^p(\mathbb{R})$, it can be approximated by a sequence functions $\{f_n\}_{n\in\mathbb{N}}\in C_0^\infty(\mathbb{R})$. I was wondering if it's possible to have the sequence $\{f_n\}_{n\in\mathbb{N}}$ satisfy a prescribed pointwise bound, such as $$f_n\leq Mf;\;\;M>0$$ almost everywhere (w.r.t. Lebesgue measure) for some $M$. This inequation means that the approximating $f_n$'s become small wherever $f$ is small; and they cannot decay to $0$ slower than the original function. Intuitively this seems possible, I just don't know how to prove it or construct such a sequence. Can it be done (e.g. using a mollifier)?

Context: In this paper: https://arxiv.org/pdf/0710.1275v1.pdf, the author proves that this condition is sufficient for a sequence of non-negative integrable functions that converge almost everywhere to have converging differential entropies, where differential entropy is defined as $$\mathcal{H}(f)=\int_{\mathbb{R}}-f\log(f)\,dx.$$ I'm trying to prove the existence of solution to an initial value problem for a PDE, and so I'd like to approximate my initial datum with smooth functions, but I'd like to have a sequence in which $\mathcal{H}(f_n)$ converges to it's initial value as well.



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