Let $\Gamma = \{f\in C_c(\mathbb{R}, \mathbb{R}):\int_{[-1,1]} f\ dm = 0\}\subset \textrm{Re} L^1(\mathbb{R},m)$ where $L^1$ has the Banach space norm $||f||_1 = \int_\mathbb{R} |f|\ dm$
What is the closure $\overline{\Gamma}$ ?
Thoughts:
$C_c$ is normally dense in $L^1$
If $f\in L^1$ is zero on some set of positive measure in $[-1,1]$, then $f\in \overline{\Gamma}$
If $\int_{[-1,1]} f\ dm = 0$, then $f\in\overline{\Gamma}$
$\overline{\Gamma}\neq L^1$, since $\int_\mathbb{R} |\chi_{[-1,1]} - g|\ dm \geq 1 ,\ \forall g\in \Gamma$
Besides this, I don't know how to formulate the closure of $\Gamma$.
The closure is $\{f \in ReL^{1}:\int_{-1}^{1} f \,=0\}$. Given $f \in L^{1}$ such that $\int_{-1}^{1} f \, dm=0$ choose $\{f_n\} \subset C_c(\mathbb R)$ such that $\int |f_n-f| \, dm \to 0$. Fix a positive function $\phi \in C_c(\mathbb R)$ such that $\int_1^{1} \phi =1$ and $\phi$ has support in $(-1,1)$. Let $g_n=f_n-(\int_1^{1} f_n) \phi$. Then $g_n \in \Gamma$ for all $n$ and $g_n \to f$ in $L^{1}$ because $\int_1^{1} f_n \to \int_1^{1} f=0$.