Exercise 5, § 6

a) Let $\mu$ be Lebesgue measure on the interval $X = ]0, +\infty[$. For every number $p$ such that $0 < p \leq +\infty$, give examples of measurable functions $f \geq 0$ on $X$, such that the set of numbers $r$ ($0 \leq r \leq +\infty$), for which $N_r(f) = (\int f^r d\mu)^{1/r}$ is finite, is one of the intervals $]0, p[, ]0, p] , ]p, +\infty[ , [p, +\infty]$ (take for $f$ functions of the form $x^\alpha (\log x)^\beta$ in the neighborhood of 0 or of $+\infty$); from this, deduce that for every interval I contained in $]0, +\infty[$, there exists a measurable function $f \geq 0$ such that I is identical to the set of $r > 0$ for which $N_r(f) < +\infty$ (consider the sum of two functions for which I has one of the above four forms).

b) For Lebesgue measure on the interval $]0, 1[$, similarly give examples of functions $f$ such that the set I of numbers $r > 0$ for which $N_r(f) < +\infty$ is an arbitrary interval contained in $]0, +\infty[$ with left end-point 0 .