This paper extends a previous one [ibid. 16, No. 3, 227-239 (1990; Zbl 0708.60039)] by the same author. In the setting of a probability space \((\Omega, A, \mathbb{P})\) with an arbitrarily indexed family of sub-\(\sigma\)- fields \(\{F_ t\}_{t \in T}\), the concept of atomic Hardy spaces \(H^ q\), \(q \in (1,\infty]\), in the spirit of \textit{R. R. Coifman} and \textit{G. Weiss} [Bull. Am. Math. Soc. 83, 569-645, (1977; Zbl 0358.30023)] is introduced. For conjugate \(q,q'\) it is shown that \((H^ q)^* = \text{BMO}^ +_ q\), and \(\text{BMO}^ +_ q \subset (H^ \infty)^*\) as a subspace. Equality holds, if \(A\) is generated by a regular chain of \(\{F_ t\}_{t \in T}\). If the \(F_ t\) are generated by finitely many atoms, \(T \subset \mathbb{N}^ 2\) equipped with a cut-parameter ordering, the duality \((\text{VMO}^ +_ q)^* = H^{q'}\), \(q \in [1,\infty)\), \(q,q'\) conjugate, is proved, \(\text{VMO}^ +_ q\) being a subspace of \(\text{BMO}^ +_ q\). The proof reveals the following inequalities \[ \| f \|_{{\mathcal H}^{q'}} \sim \sum_{t \in T} \biggl \| \mathbb{E} \bigl( | f_ t |^{q'} \mid {\mathcal F}_ t \bigr)^{1/q'} \biggl \|_{L'}. \] Similar assertions hold if one considers the measure situation \((\Omega, A, \mathbb{P}) = ([0,1), B([0,1)),dx)\) with the one-dimensional Walsh system as index set \(T\) (ordered by inclusion) and \(F_ t = \sigma\) (Walsh functions on \(t)\) (`tree case generated by the Walsh system').