Consider one-dimensional Lebesgue measure on ([0,1]). Verify that the sequence (u_{n}(x):=) (n 1_{(0,1 / n)}(x), n in
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Consider one-dimensional Lebesgue measure on \([0,1]\). Verify that the sequence \(u_{n}(x):=\) \(n 1_{(0,1 / n)}(x), n \in \mathbb{N}\), converges pointwise to the function \(u \equiv 0\), but that no subsequence of \(u_{n}\) converges in \(\mathcal{L}^{p}\)-sense for any \(p \geqslant 1\).
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That for every fixed x the sequence ux n1l 01nx is obvio...View the full answer
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