Geodesic
A Note on Epi-Convergence
Canadian mathematical bulletin, Tome 37 (1994) no. 3, pp. 294-300

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DOI

Let LSC(X) denote the set of extended real valued lower semicontinuous functions on a metrizable space X. If f, f1, f2, f3 ,... is a sequence in LSC(X), we say 〈fn〉 is epi-convergent to f provided the sequence of epigraphs 〈epi fn 〉 is Kuratowski- Painlevé convergent to epi f. In this note we address the following question: what conditions on f and/or on X are necessary and sufficient for this mode of convergence to force epigraphical convergence with respect to the stronger Hausdorff metric and Vietoris topologies?
DOI : 10.4153/CMB-1994-044-7
Mots-clés : Primary: 54B20, 26A15, secondary: 54C35, epi-convergence, lower semicontinuous function, Kuratowski-Painlevé convergence, Fell topology, Hausdorff distance, Vietoris topology 
Beer, Gerald. A Note on Epi-Convergence. Canadian mathematical bulletin, Tome 37 (1994) no. 3, pp. 294-300. doi: 10.4153/CMB-1994-044-7
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     title = {A {Note} on {Epi-Convergence}},
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     year = {1994},
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     doi = {10.4153/CMB-1994-044-7},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1994-044-7/}
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