  Let $L_{q,\mu},\, 1\le q<\infty, \ \mu\ge0,$ denote the weighted $L_q$ space
with the classical Jacobi weight $w_\mu$ on the ball $\Bbb B^d$. We consider
the weighted least $\ell_q$ approximation problem for a given
$L_{q,\mu}$-Marcinkiewicz-Zygmund family on $\Bbb B^d$. We obtain the weighted
least $\ell_q$ approximation errors for the weighted Sobolev space
$W_{q,\mu}^r$, $r>(d+2\mu)/q$, which are order optimal. We also discuss the
least squares quadrature induced by an $L_{2,\mu}$-Marcinkiewicz-Zygmund
family, and get the quadrature errors for $W_{2,\mu}^r$, $r>(d+2\mu)/2$, which
are also order optimal. Meanwhile, we give the corresponding the weighted least
$\ell_q$ approximation theorem and the least squares quadrature errors on the
sphere.
