Exercise 1.5.15 (Uniformly integrable but not convergent sequences)

Question

Give an example of uniformly integrable functions that converge pointwise a.e. to zero, but do not converge in measure.

Elu Thingol · Accepted Answer

The only case not covered by the restrictions of the uniform integrability is the escape to horizontal infinity. Let $f_n:=1_{[n,n+1]}$. Then it is easy to verify that $\left(f\right)_{n\in\mathbb{N}}$ is uniformly integrable. From Example 1.5.2 we now that $\left(f\right)_{n\in\mathbb{N}}$ converges pointwise to zero, but not uniformly, almost uniformly, in $L^1$ norm nor in measure.

Exercise 1.5.15 (Uniformly integrable but not convergent sequences)

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Exercise 1.5.15 (Uniformly integrable but not convergent sequences)

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