Examples For Strict Inequality
Equip the space with the Borel σ-algebra and the Lebesgue measure.
- Example for a probability space: Let denote the unit interval. For every natural number define
- Example with uniform convergence: Let denote the set of all real numbers. Define
These sequences converge on pointwise (respectively uniformly) to the zero function (with zero integral), but every has integral one.
Read more about this topic: Fatou's Lemma
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