  Most of the popular dependence measures for two random variables $X$ and $Y$
(such as Pearson's and Spearman's correlation, Kendall's $\tau$ and Gini's
$\gamma$) vanish whenever $X$ and $Y$ are independent. However, neither does a
vanishing dependence measure necessarily imply independence, nor does a measure
equal to 1 imply that one variable is a measurable function of the other. Yet,
both properties are natural properties for a convincing dependence measure. In
this paper, we present a general approach to transforming a given dependence
measure into a new one which exactly characterizes independence as well as
functional dependence. Our approach uses the concept of monotone rearrangements
as introduced by Hardy and Littlewood and is applicable to a broad class of
measures. In particular, we are able to define a rearranged Spearman's $\rho$
and a rearranged Kendall's $\tau$ which do attain the value $0$ if and only if
both variables are independent, and the value $1$ if and only if one variable
is a measurable function of the other. We also present simple estimators for
the rearranged dependence measures, prove their consistency and illustrate
their finite sample properties by means of a simulation study and a data
example.
