Kudō [Ann. Probab. 2(1) (1974), 76-83] introduced a notion of convergence of sub-σ-fields in terms of L 1 -convergence of conditional expectation. We study relationships between convergence of sub-σ-fields and convergence of associated subspaces L p n . In particular, we show that the Slice convergence is not adapted to the L 1 -case in contrast to the Mosco convergence. Notions of upper and lower limits are also studied. This enables us to obtain measurability results regarding those limits. We also give examples and counterexamples in order to exhibit characteristic behaviours of the sequences n n , L p n n , and Ɛ n n .

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In the context of graph layout, many algorithms have been designed to remove node overlapping, and many quality criteria and associated metrics have been proposed to evaluate those algorithms. Unfortunately, a complete comparison of the algorithms based on some metrics that evaluate their quality has never been provided and it is thus difficult for a visualisation designer to select the algorithm that best suits their needs. In this paper, we review 22 metrics available in the literature, classify them according to the quality criteria they try to capture, and select a representative one for each class. Based on the selected metrics,we compare 9 node overlap removal algorithms. Our experiment involves 854 synthetic and real-world graphs. Finally, we propose a JavaScript library containing both the algorithms and the criteria, and we provide a Web platform, AGORA, in which one can upload graphs, apply the algorithms and compare/download the results.

Many algorithms have been designed to remove node overlapping, and many quality criteria and associated metrics have been proposed to evaluate those algorithms. Unfortunately, a complete comparison of the algorithms based on some metrics that evaluate the quality has never been provided and it is thus difficult for a visualization designer to select the algorithm that best suits his needs. In this paper, we review 21 metrics available in the literature, classify them according to the quality criteria they try to capture, and select a representative one for each class. Based on the selected metrics, we compare 8 node overlap removal algorithms. Our experiment involves 854 synthetic and real-world graphs.

H. Kudō introduced a notion of convergence of a sequence Bn of sub-σ-fields. We emphasize a connection with Mosco-convergence of associated subspaces L p E (Bn) where E is a separable Banach space with separable dual. This enables us to study the convergence of conditional expectation sequences of random sets and therefore to obtain a new version of the multivalued Fatou lemma.1991 Mathematics Subject Classification. 28B20.