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Measure and Integration Theory
Published in Athanasios Christou Micheas, Theory of Stochastic Objects, 2018
If we replace condition (ii) of definition 3.11 with (ii)′ μ(A)∈[−∞,0],∀A∈A, then μ is called a signed nonpositive measure. Clearly, a measure is a special case of a signed measure, however a signed measure is not in general a measure. We say that a set A is a positive set with respect to a signed measure μ if A is measurable and for every measurable subset of E of A we have μ(E) ≥ 0. Every measurable subset of a positive set is again positive and the restriction of μ to a positive set yields a measure.
Basic Results on Measure and Integration
Published in Fabio Silva Botelho, Functional Analysis, Calculus of Variations and Numerical Methods for Models in Physics and Engineering, 2020
Definition 7.5.2 (Signed measure)Let (Uℳ) be a measurable space. We say that v: ℳ → [−∞,+∞] is a signed measure ifv may assume at most one the values −∞,+∞.v (ø) = 0.v (∑∞n=1En) = ∑∞n=1 v(En) for all sequence of measurable disjoint sets {En}
Systemic risk in a mean-field model of interbank lending with self-exciting shocks
Published in IISE Transactions, 2018
Anastasia Borovykh, Andrea Pascucci, Stefano La Rovere
Consider again the model defined in Equation (3). In order to improve the first-order approximation of νMt given in Equation (17), we can analyze the fluctuations of νM around its large system limit ν. Following Spiliopoulos et al. (2014) define The signed-measure-valued process ΞM weakly converges to the fluctuation limit in an appropriate space (in particular the convergence is considered in weighted Sobolev spaces, in which the sequence ΞM, can be shown to be relatively compact; for discussion on this space, as well as the existence and uniqueness of the limiting point, we refer to Sections 7, 8, and 9 in Spiliopoulos et al. (2014)). We start by deriving an expression for ΞMt. Some terms in this expression will vanish in the limit of M → ∞, and using the tightness of the processes (see Section 8 in Spiliopoulos et al., 2014) and continuity of the operators in the expression for ΞM we can pass to the limit and find the expression that the limiting fluctuation process satisfies.