Exercise 29, § 5

Let μ be a measure on a locally compact space X. For every μ-measurable numerical function f ≥ 0 defined on X, the mapping f'_μ : t ↦ |\mu|^*(f([t, +∞])) of $\mathbf{R}_+$ into $\overline{\mathbf{R}}_+$ is decreasing and right-continuous. If $\nu$ is a measure on a second locally compact space Y and g is a $\nu$-measurable numerical function $\geqslant 0$ defined on Y, f and g are said to be equimeasurable (for $\mu$ and $\nu$ respectively) if $f'_\mu = g'_\nu$. One denotes by $f^*$ (or $f^*_\mu$) the function defined on $\mathbf{R}_+$ that is equal, for all $s \in \mathbf{R}_+$, to the supremum in $\overline{\mathbf{R}}_+$ of the set of numbers a such that $f'_\mu(a) \geqslant s$ (the supremum being equal to 0 if this set is empty).

a) Show that the function $f^*$ is decreasing, left-continuous in $\mathbf{R}_+$, and equimeasurable to f (for the measure $\mu$ and Lebesgue measure on $\mathbf{R}_+$); $f^*$ is called the decreasing rearrangement of f.

b) If $0 \leqslant f \leqslant g$ are two $\mu$-measurable functions on X, show that $f^* \leqslant g^*$. If $(f_n)$ is an increasing sequence of $\mu$-measurable functions $\geqslant 0$ on X, and $f = \sup_n f_n$, show that $f^*(s) = \sup_n f^*_n(s)$ at all the points where $f^*$ is continuous.

c) Show that for every $\mu$-measurable numerical function $f \geqslant 0$, $|\mu|^*(f) = \int^* f^*(s) \, ds$. (Consider first the case of a step function, then make use of b).)

d) Let w be a numerical function defined on $\mathbf{R}_+$ that is finite (except possibly at the point $s = 0$) and decreasing. For every $\mu$-measurable numerical function $f \geqslant 0$ defined on X, set

Show that this function satisfies the conditions of Exer. 28. (To prove that it is convex, consider first the case that w is a decreasing step function, using c), then pass to the limit to treat the general case.)

e) With the notations of d), write $\mathscr{L}_F^{p,w}$ instead of $\mathscr{L}_F^\lambda$, and set $N_{p,w}(f) = \lambda(|f|)$ for every $\mu$-measurable mapping f of X into F. Show that if $f \in \mathscr{L}_F^{p,w}$ and if, for every n, $f_n$ denotes the function equal to f if $|f| \leqslant n$ and to $nf/|f|$ if $|f| > n$, then the sequence $(|f_n|^*)$ tends almost everywhere to $|f|^*$ and one has $\lim_{n \to \infty} N_{p,w}(f - f_n) = 0$.

If f is $\mu$-negligible, then $N_{p,w}(f) = 0$.