MathLiveFatou's lemma
For a sequence of non-negative measurable functions, the integral of the limit inferior is at most the limit inferior of the integrals. It is used to bound integrals of limits when the full convergence needed for dominated or monotone convergence is not available.
\int_X \left(\liminf_{n\to\infty} f_n\right)\,\mathrm{d}\mu \le \liminf_{n\to\infty} \int_X f_n\,\mathrm{d}\muWhat the symbols mean
| the measure space on which the functions are defined | |
| the measure on X | |
| a sequence of non-negative measurable functions on X | |
| the pointwise limit inferior of the sequence f_n | |
| integration with respect to the measure \mu |
About this formula
- FieldMathematics
- TopicMeasure theory and integration
- LevelUniversity
- Also calledFatou inequality, liminf integral inequality, integral of liminf inequality, Lebesgue integral limit inferior inequality
- SourceWikidata Q1068118
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