  We study tent spaces on general measure spaces $(\Omega, \mu)$. We assume
that there exists a semigroup of positive operators on $L^p(\Omega, \mu)$
satisfying a monotone property but do not assume any geometric/metric structure
on $\Omega$. The semigroup plays the same role as integrals on cones and cubes
in Euclidean spaces. We then study BMO spaces on general measure spaces and get
an analogue of Fefferman's $H^1$-BMO duality theory. We also get a $H^1$-BMO
duality inequality without assuming the monotone property.
  All the results are proved in a more general setting, namely for
noncommutative $L^p$ spaces.
