Let $X$ be a metric space, $\mu$ a $\sigma$-finite non-negative Borel measure on $X$, and $(E, |\cdot|)$ a Banach space. Let $\mathcal L_p := \mathcal L_p (X, \mu, E)$ and $\|\cdot\|_{\mathcal L_p}$ be its semi-norm. Here we use Bochner integral. Let $\mathcal C_c :=\mathcal C_c(X, E)$ be the space of all $E$-valued continuous functions on $X$ with compact supports. It is well-known that

Theorem If $X$ is locally compact separable, then $\mathcal C_c$ is dense in $\big (\mathcal L_p, \|\cdot\|_{\mathcal L_p} \big)$ for all $p \in [1, \infty)$.

I would like to ask if above result can be further strengthened, i.e.,

Let $X$ be locally compact separable. For each $f \in \mathcal L_p$ with $p \in [1, \infty)$ and $\varepsilon>0$, there is $g \in \mathcal C_c$ such that $$ \color{red}{|g| \le |f|} \quad \mu\text{-a.e.} \quad \quad \text{and} \quad \quad \| f-g \|_{\mathcal L_p} < \varepsilon. $$

This result, if true, generalizes this lemma which itself generalizes another lemma.

Thank you so much for your elaboration!

2more comments