It is clear that any measurable function can be approximated pointwise boundedly by a sequence of simple functions, and it is not hard to understand that any integrable function can be approximated in $L^p$ by continuous functions with finite support.
However, when the conditions are combined together, I wonder if the following statement is true:
Let $f$ be a non-negative measurable function and suppose that $f\in L^p(\Bbb R^n)$, then for any $\delta \gt 0$ there exists a non-negative continuous function with compact support, note as $h_\delta$, such that
(1) $\int |f(x)|^p dx - \delta^p \le \int|h_\delta(x)|^pdx$ and
(2) $h_\delta(x)\le f(x)$ for a.e. $x\in \Bbb R^n$.
Could anyone help me by giving either a direct proof or a counterexample to the statement? Thanks a lot.