= Density of bounded centered scores
{title2=$\overline{\{g\text{ bounded}:Pg=0\}}^{L^2(P)}=L^2_0(P)$}
For $h\in L^2_0(P)$, truncate $h$ to $h_m=\max(-m,\min(h,m))$ and subtract $Ph_m$. <Dominated convergence> gives $h_m\to h$ in <L2 space>, and the <Cauchy-Schwarz inequality> gives $Ph_m\to0$. 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.
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