As in the title, I want to know if compactly supported bounded functions with zero mean, i.e. $\int f dx= 0$, are dense in $L^p$ for any $ 1 <p< \infty $. I feel like there should be some counterexamples to this but could not think of one.
Thanks in advance.
Take $f \in L^p(\mathbb R^n),$ where $p\in (1,\infty)$. Let $\epsilon > 0$. Then there is $g \in C^\infty_c (\mathbb R^n)$ such that $$\| f - g \|_{L^p(\mathbb R^n)} < \epsilon.$$ Define the constant $$C = \int_{\mathbb R^n} g(x) dx,$$ and define a sequence $$g_k(x) = g(x) - \frac{C}{(2k)^n} \chi_{[-k,k]^n}(x),$$ where $\chi_{[-k,k]^n}$ is the indicator function of the hypercube $[-k,k]^n$. Notice that $$\int_{\mathbb R^n} g_k(x) dx = C - \frac{C}{(2k)^n} \text{Vol}([-k,k]^n) = 0.$$ Thus $g_k$ is a bounded function with compact support and $\int_{\mathbb R^n} g_k(x) dx = 0$. Consider, $$\| f - g_k \|_{L^p(\mathbb R^n)} \le \| f - g\|_{L^p(\mathbb R^n)} + \frac{C}{(2k)^n} \| \chi_{[-k,k]^n}\|_{L^p(\mathbb R^n)}$$ and $$\| \chi_{[-k,k]^n}\|_{L^p(\mathbb R^n)} = (2k)^{n/p}.$$ This shows that $$\|f - g_k\|_{L^p(\mathbb R^n)} \le \underbrace{\| f -g \|_{L^p(\mathbb R^n)}}_{\,\,\,\,\,\,\,< \epsilon} + \underbrace{\frac{C}{(2k)^{n(1-1/p)}}}_{\,\,\,\to 0 \text{ as } k \to \infty}. $$ Thus for $k$ sufficiently large, $$\| f - g_k \|_{L^p(\mathbb R^n)} < 2 \epsilon.$$ Hence, $f$ can be approximated arbitrarily well in $L^p(\mathbb R^n)$ by a bounded, compactly supported function which integrates to zero.
