Fatou's Lemma - Examples For Strict Inequality

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

$f_n(x)=\begin{cases}n&\text{for }x\in (0,1/n),\\ 0&\text{otherwise.} \end{cases}$

Example with uniform convergence: Let denote the set of all real numbers. Define

$f_n(x)=\begin{cases}\frac1n&\text{for }x\in ,\\ 0&\text{otherwise.} \end{cases}$

These sequences converge on pointwise (respectively uniformly) to the zero function (with zero integral), but every has integral one.

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