Let $A\subset \mathbb{R}$ be measurable such that there are $a,b\in \mathbb{R}$, $a<b$ fulfilling $[b,\infty)\subset A\subset [a,\infty)$. The right rearrangement of $A^{\*}$ of $A$ is defined as $A^{\*}=[c,\infty)$ where $c:=b-|A\cap [a,b]|$. Now we can define the increasing rearrangement as follows: For a function $u:\mathbb{R}\to[0,1]$ fulfilling $\lim\_{x\to -\infty} u(x)=0$ and $\lim\_{x\to \infty} u(x)=1$, let the increasing rearrangement $u^{\*}:\mathbb{R}\to [0,1]$ be the function which fulfils for all $t\in (0,1)$
\begin{align\*}
\{x\in\mathbb{R}: u^{\*}(x)\geq t\}=\{x\in \mathbb{R}: u(x)\geq t\}^{\*}.
\end{align\*}
To show: Assume that $(u\_n)\_{n\in \mathbb{N}}:\mathbb{R}\to [0,1]$ is a sequence which converges pointwise a.e. to $u$. Then the sequence of increasing rearrangements $(u\_n^{\*})\_{n\in \mathbb{N}}$ is converging a.e. to $u^{\*}$.



Convergence of increasing rearrangment
How about $u\_n = 1\_{[-n-1, -n] \cup [1,\infty)}$ (the characteristic function of $[-n-1, -n] \cup [1,\infty)$)? Then $u\_n^\* = 1\_{[0,\infty)}$ for all $n$ but $u\_n$ converges pointwise to $u = 1\_{[1,\infty)}$, which has $u^\* = 1\_{[1,\infty)}$.


