  Given a sequence of Marcinkiewicz-Zygmund inequalities in $L_2$ on a compact
space, Gr\"ochenig in \cite{G} discussed weighted least squares approximation
and least squares quadrature. Inspired by this work, for all $1\le p\le\infty$,
we develop weighted least $\ell_p$ approximation induced by a sequence of
Marcinkiewicz-Zygmund inequalities in $L_p$ on a compact smooth Riemannian
manifold $\Bbb M$ with normalized Riemannian measure (typical examples are the
torus and the sphere). In this paper we derive corresponding approximation
theorems with the error measured in $L_q,\,1\le q\le\infty$, and least
quadrature errors for both Sobolev spaces $H_p^r(\Bbb M), \, r>d/p$ generated
by eigenfunctions associated with the Laplace-Beltrami operator and Besov
spaces $B_{p,\tau}^r(\Bbb M),\, 0<\tau\le \infty, r>d/p $ defined by best
polynomial approximation. Finally, we discuss the optimality of the obtained
results by giving sharp estimates of sampling numbers and optimal quadrature
errors for the aforementioned spaces.
