Abstract
We show that a probability measure on a metric space has full support, if, and only if, the set of all probability measures, that are absolutely continuous with respect to it, is dense in the set of all Borel probability measures. We illustrate the result through a general version of Laplace’s method, which in turn leads to general stochastic convergence to global maxima.
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\(\int _{X\setminus \left\{ x^{u}\right\} }\left( u\left( x^{u}\right) -u\left( x\right) \right) d\mu \left( x\right) =0\) would imply \(\mu \left( \left\{ x\in X\setminus \left\{ x^{u}\right\} :u\left( x^{u}\right) -u\left( x\right) >0\right\} \right) =0\), a contradiction because \(u\left( x^{u}\right) -u\left( x\right) >0\) for all \(x\in X\setminus \left\{ x^{u}\right\} \); thus, \(\left\{ x\in X\setminus \left\{ x^{u}\right\} :u\left( x^{u}\right) -u\left( x\right) >0\right\} =X\setminus \left\{ x^{u}\right\} \).
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Acknowledgements
We thank Giacomo Cattelan, Ludovica Ciasullo, and Isabella Morgan Wolfskeil for excellent research assistance. Simone Cerreia-Vioglio, and Fabio Maccheroni and Massimo Marinacci gratefully acknowledge the financial support of ERC (Grants SDDM-TEA and INDIMACRO, respectively).
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Communicated by Nizar Touzi.
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Cerreia-Vioglio, S., Maccheroni, F. & Marinacci, M. A Characterization of Probabilities with Full Support and the Laplace Method. J Optim Theory Appl 181, 470–478 (2019). https://doi.org/10.1007/s10957-018-01459-7
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DOI: https://doi.org/10.1007/s10957-018-01459-7