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Register a new userFatous Lemma in Its Classic Form and Lebesgues Convergence Theorems for Varying Measures with Applications to MDPs
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The classic Fatou lemma states that the lower limit of a sequence of integrals of functions is greater or equal than the integral of the lower limit. It is known that Fatous lemma for a sequence of weakly converging measures states a weaker inequality because the integral of the lower limit is replaced with the integral of the lower limit in two parameters, where the second parameter is the argument of the functions. This paper provides sufficient conditions when Fatous lemma holds in its classic form for a sequence of weakly converging measures. The functions can take both positive and negative values. The paper also provides similar results for sequences of setwise converging measures. It also provides Lebesgues and monotone convergence theorems for sequences of weakly and setwise converging measures. The obtained results are used to prove broad sufficient conditions for the validity of optimality equations for average-cost Markov decision processes.
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The risk of extreme environmental events is of great importance for both the authorities and the insurance industry. This paper concerns risk measures in a spatial setting, in order to introduce the spatial features of damages stemming from environmental events into the measure of the risk. We develop a new concept of spatial risk measure, based on the spatially aggregated loss over the region of interest, and propose an adapted set of axioms for these spatial risk measures. These axioms quantify the sensitivity of the risk measure with respect to the space and are especially linked to spatial diversification. The proposed model for the cost underlying our definition of spatial risk measure involves applying a damage function to the environmental variable considered. We build and theoretically study concrete examples of spatial risk measures based on the indicator function of max-stable processes exceeding a given threshold. Some interpretations in terms of insurance are provided.
We prove the Zabreikos lemma in 2-Banach spaces. As an application we shall prove a version of the closed graph theorem and open mapping theorem.
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