English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The classic Fatou lemma states that the lower limit of a sequence ofintegrals of functions is greater or equal than the integral of the lowerlimit. It is known that Fatou's lemma for a sequence of weakly convergingmeasures states a weaker inequality because the integral of the lower limit isreplaced with the integral of the lower limit in two parameters, where thesecond parameter is the argument of the functions. This paper providessufficient conditions when Fatou's lemma holds in its classic form for asequence of weakly converging measures. The functions can take both positiveand negative values. The paper also provides similar results for sequences ofsetwise converging measures. It also provides Lebesgue's and monotoneconvergence theorems for sequences of weakly and setwise converging measures.The obtained results are used to prove broad sufficient conditions for thevalidity of optimality equations for average-cost Markov decision processes.

Fatou's Lemma in Its Classic Form and Lebesgue's Convergence Theorems for Varying Measures with Applications to MDPs