Existence of probability measures with given marginals (Q1180586)

The authors discuss the following problem: Let \(\lambda\) be a finite measure, let \(0\leq f\leq 1\) be a measurable real function and let \(\pi_ j\), \(1\leq j\leq N\), be \(N\) measurable functions. Under which conditions on \(\lambda\) can one find a measurable function \(g\) with values only 0 and 1 such that the distributions of \(\pi_ j\), \(1\leq j\leq N\), are the same for the measures with density \(f\) and with density \(g\)? The case of linear projections on \(\mathbb{R}^ p\) has a relation to a problem in tomography.