Good filtrations and $F$-purity of invariant subrings

Let $k$ be an algebraically closed field of positive characteristic, $G$ a reductive group over $k$, and $V$ a finite dimensional $G$-module. Let $B$ be a Borel subgroup of $G$, and $U$ its unipotent radical. We prove that if $S=\Sym V$hasagoodfiltration,then$S^U$is$F$-pure.