Dirac measure

Dirac measure

Measurable Sets

Published in Hugo D. Junghenn, Principles of Analysis, 2018

( $ \boldsymbol{(} $ d) $ \boldsymbol{)} $ Dirac measure. Let (X,F) $ (X,\boldsymbol{{ \fancyscript {F}}}) $ be a measurable space. For x∈X $ x \in X $ and A∈F $ A \in \boldsymbol{{ \fancyscript {F}}} $ define δx(A)=1A(x) $ \delta _x(A)= {\boldsymbol{1}}_A(x) $ . Then δx $ \delta _x $ is a probability measure on F $ \boldsymbol{{ \fancyscript {F}}} $ .

Preference robust models in multivariate utility-based shortfall risk minimization

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Journal Information

Published in Optimization Methods and Software, 2022

Yuan Zhang, Huifu Xu, Wei Wang

By the triangle inequality, we have where denotes the Dirac measure at . It suffices to show existence of appropriate positive numbers δ and such that for with , we have and for all . By Strassen's theorem [33, Theorem 2.13], (62) and (63) are implied respectively by and Note that (64) follows from Theorem 4.9. Thus we are left to prove (65). Note that It follows by Theorem 4.7 that we can choose δ sufficiently small such that when , we have Analogous to (64), we have from Theorem 4.9 for all . Combining (66) and (67), we have (65) holds.

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Measurable Sets

Preference robust models in multivariate utility-based shortfall risk minimization