Let ( X , A , μ ) (X,\mathfrak {A},\mu ) be a σ \sigma -finite measure space and K \mathcal {K} be a linear subspace of L 1 ( μ ) {\mathcal {L}_1}(\mu ) with K = X \mathcal {K} = X . The following inverse problem is treated: Which sets A ∈ A A \in \mathfrak {A} are " K \mathcal {K} -determined" within the class of all functions g ∈ L ∞ ( μ ) g \in {\mathcal {L}_\infty }(\mu ) satisfying 0 ≤ g ≤ 1 0 \leq g \leq 1 , i.e. when is g = 1 A g = {1_A} the unique solution of ∫ f g d μ = ∫ f 1 A d μ \smallint fg\;d\mu = \smallint f{1_A}\;d\mu , f ∈ K ? f \in \mathcal {K}? Recent results of Fishburn et al. and Kemperman show that the condition A = { f ≥ 0 } A = \{ f \geq 0\} for some f ∈ K f \in \mathcal {K} is sufficient but not necessary for uniqueness. To obtain a complete characterization of all K \mathcal {K} -determined sets, K \mathcal {K} has to be enlarged to some hull K ∗ {\mathcal {K}^{\ast } } by extending the usual weak convergence to limits not in L 1 ( μ ) {\mathcal {L}_1}(\mu ) . Then one of the main results states that A A is K \mathcal {K} -determined if and only if there is a representation A = { f ∗ > 0 } A = \{ {f^{\ast } } > 0\} and X ∖ A = { f ∗ > 0 } X\backslash A = \{ {f^{\ast } } > 0\} for some f ∗ ∈ K ∗ {f^{\ast }} \in {\mathcal {K}^{\ast } } .