Density of bounded centered scores (${\overline{\{g\text{ bounded}:Pg=0\}}}^{L^2(P)}=L_0^2(P)$)

For $h\in L_0^2(P)$, truncate $h$ to $h_m=\max (-m,\min (h,m))$ and subtract $P{h}_m$. Dominated convergence gives $h_m\to h$ in L2 space, and the Cauchy-Schwarz inequality gives $P{h}_m\to 0$. Thus bounded centered directions are dense in the mean-zero L2 space. The measure defining the L2 norm is essential: density-weighted L2 space can contain functions outside the unweighted Lebesgue space.