Etichetă: Organisation Internationale de la Francophonie
Mesures vectorielles – Etudes & Recherches (Integral International Institute of...
Une mesure vectorielle est une généralisation de la notion de mesure classique. Au lieu de prendre des valeurs dans les nombres réels positifs (mesure positive) ou dans les nombres réels (mesure signée), elle prend des valeurs dans un espace vectoriel normé. Cela signifie que chaque ensemble mesurable se voit attribuer un vecteur d'un espace vectoriel.
Exemples. Mesure à valeurs dans un espace de Banach. On peut définir une mesure dont les valeurs sont des éléments d'un espace de Banach, comme l'espace des opérateurs linéaires sur un espace de Hilbert.
Mesure vectorielle et partitions continues de l'unité. Ce concept est utilisé dans l'étude de certaines mesures vectorielles et de leurs propriétés.
Une mesure vectorielle est une généralisation de la notion de mesure abstraite: c'est une mesure à valeurs dans un espace vectoriel ad hoc (au lieu de‾R+ pour une mesure positive, ou de‾R pour une mesure signée).
Note. La théorie de la mesure est la branche des mathématiques qui traite des espaces mesurés et est le fondement axiomatique de la théorie des probabilités. En 1894, Émile Borel énonce la première définition d'ensemble négligeable. En 1897, il définit les ensembles mesurables. En 1901, Henri-Léon Lebesgue introduit la notion de mesure. La théorie se développe jusque dans les années 1950. Andreï Kolmogorov proposera une axiomatisation du calcul des probabilités basée notamment sur l'intégrale définie à partir d'une mesure. Lebesgue et ses successeurs ont été amenés à généraliser la notion d'intégrale au point d'en faire ce que certains appellent une intégrale abstraite. L'aire sous une courbe est calculée par une somme de petits rectangles dont la hauteur représente la valeur moyenne de la fonction sur un intervalle et la base représente la mesure de l'intervalle. Sur une droite réelle, la mesure de Lebesgue d'un intervalle est la différence des distances par rapport à l'origine. Mais une mesure est une fonction, et cela a donc amené les mathématiciens de l'époque à généraliser l'intégrale non plus selon une mesure particulière, celle de Lebesgue, mais selon n'importe quelle mesure. C'est comme si pour mesurer un intervalle on utilisait un abaque (mesures discrètes) ou tout autre instrument plutôt qu'un mètre-ruban.
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