Ivan Shishkin, Birch Grove

Théorème de convergence montone

Definition / Functional analysis /

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Définition formelle

Soit (E,T,μ)(E,T,\mu) un espace mesuré.
Soit (fn)n(f_{n})_{n} une suite croissante de fonctions mesurables positives.
Alors
lim⁡n→+∞∫Efndμ=∫Elim⁡n→+∞fndμ.\lim_{n\to + \infty} \int_{E} f_{n} d\mu = \int_{E} \lim_{n \to + \infty}f_{n} d\mu .

Practice this concept with exercises

  • Dans l'espace mesuré ([0,1],B([0,1]),λ)([0,1],\mathcal B([0,1]),\lambda), calculer
    lim⁡n→+∞∫0111+xn dx.\lim_{n\to+\infty} \int_0^1 \frac{1}{1+x^n}\,dx.

    Open exercise
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