Consider the functions (i) (u(x)=frac{1}{x}, quad x in[1, infty)); (ii) (v(x)=frac{1}{x^{2}}, quad x in[1, infty)); (iii) (quad
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Consider the functions
(i) \(u(x)=\frac{1}{x}, \quad x \in[1, \infty)\);
(ii) \(v(x)=\frac{1}{x^{2}}, \quad x \in[1, \infty)\);
(iii) \(\quad w(x)=\frac{1}{\sqrt{x}}, \quad x \in(0,1]\)
(iv) \(y(x)=\frac{1}{x}, \quad x \in(0,1]\)
and check whether they are Lebesgue integrable in the regions given - what would happen if we consider \(\left[\frac{1}{2}, 2ight]\) instead?
[ consider first \(u_{k}=u \mathbb{1}_{[1, k]}\), resp., \(w_{k}=w \mathbb{1}_{[1 / k, 1]}\), etc. and use monotone convergence and the fact that Riemann and Lebesgue integrals coincide if both exist.]
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