  We prove that if $X$ is a real rearrangement-invariant function space on
$[0,1]$, which is not isometrically isomorphic to $L_2,$ then every surjective
isometry $T:X\to X$ is of the form $Tf(s)=a(s)f(\sigma(s))$ for a Borel
function $a$ and an invertible Borel map $\sigma:[0,1] \to [0,1].$ If $X$ is
not equal to $L_p$, up to renorming, for some $1\le p\le \infty$ then in
addition $|a|=1$ a.e. and $\sigma$ must be measure-preserving.

Functional Analysis