  Given a sequence of Marcinkiewicz-Zygmund inequalities in $L_2$ on a usual
compact space $\mathcal M$, Gr\"ochenig introduced the weighted least squares
polynomials and the least squares quadrature from pointwise samples of a
function, and obtained approximation theorems and quadrature errors. In this
paper we obtain approximation theorems and quadrature errors on the sphere
which are optimal. We also give upper bounds of the operator norms of the
weighted least squares operators.
