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Fatou's lemma strict inequality举了两个例子:
- Example for a probability space: Let $ S=[0,1] $ denote the unit interval. For every natural number $ n $ define $ f_{n}(x)={\begin{cases}n&{\text{for }}x\in (0,1/n),\\0&{\text{otherwise.}}\end{cases}} $
- Example with uniform convergence: Let $ S $ denote the set of all real numbers. Define $ f_{n}(x)={\begin{cases}{\frac {1}{n}}&{\text{for }}x\in [0,n],\\0&{\text{otherwise.}}\end{cases}} $
如果同时要求probability space + uniform convergence 则一定取等吗
probability space 是指全空间的测度为1. |
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