Given a pseudo-effective divisor L we construct the diminished ideal of L, a
"continuous" extension of the asymptotic multiplier ideal for big divisors to
the pseudo-effective boundary. For most pseudo-effective divisors L the
multiplier ideal of the metric of minimal singularities of L is contained in
the diminished ideal. We also characterize abundant divisors using the
diminished ideal, indicating that in this case the geometric and analytic
information should coincide.Comment: 27 pages; v3: corrected several mistake
