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.]