Let (X,\({\mathcal F},g)\) be a fuzzy measure space, as defined by \textit{M. Sugeno} in his Thesis (1974). Given any measurable function f: \(X\to {\mathbb{R}}^+_ 0\), and any set \(A\in {\mathcal F}\), the fuzzy integral of f over A, with respect to g, is the following number: \[ \int_{A}f\cdot g=\sup_{\alpha \geq 0}[\alpha \wedge g(A\cap F_{\alpha})], \] where \(F_{\alpha}=\{x\in X:\) f(x)\(\geq \alpha \}.\) The main theorem may be formulated as follows: ``Given two fuzzy measures, g and \(\gamma\), \(\gamma\leq g\), a Radon- Nikodým derivative \(h=d\gamma /dg\) does exist (with respect to the fuzzy integral above) if and only if there exists a decreasing family \((A_{\alpha})_{\alpha \geq 0},\quad A_{\alpha}\in {\mathcal F},\) satisfying \[ (*)\quad \gamma (E\cap A_{\alpha})\geq \alpha \wedge g(E\cap A_{\alpha})\quad and\quad \gamma (E)\leq g(E\cap A_{\gamma (E)}) \] for all \(E\in {\mathcal F}\), \(\alpha\geq 0.''\) This theorem is more general than Sugeno's; moreover, both of conditions in (*) are easily satisfied when X is a Polish space, \({\mathcal F}\) is the Borel \(\sigma\)-field of X, and \(\gamma\) is of ``type \(\vee ''\), i.e. \(\gamma (A\cup E)=\gamma (A)\vee \gamma (E),\) for all \(A,E\in {\mathcal F}.\) A conditioning theory follows, for measures of type \(\vee\), similar to the classical one.