Sometime back I was asked this question by our professor teaching us probability : How do we characterize random variables which have uniform distribution?. I revisited my old measure notes to give a partial answer to this question:
Let 
be a random variable defined on the probability space ![{[0,1]}](http://s0.wp.com/latex.php?latex=%7B%5B0%2C1%5D%7D&bg=ffffff&fg=000000&s=-2 "{[0,1]}")
with Lebesgue measure. Suppose further that is the union of disjoint intervals 
where in each interval 
we have either 
or 
a.e. Then, a necessary condition for to have the uniform distribution is the following. For each ![{y \in X([0,1])}](http://s0.wp.com/latex.php?latex=%7By+%5Cin+X%28%5B0%2C1%5D%29%7D&bg=ffffff&fg=000000&s=-2 "{y \in X([0,1])}")
we have
a.e (
).
Proof: Let 
, then if 
is the probability measure (lebesgue measure) on . Let 
be the intnervals where 
a.e or 
. Then,
![\displaystyle \mu X^{-1}(A) = \mu(A) = \int_A d\mu = \int_{[0,1]} \sum_{k \geq 1} {\chi_{I_k \cap A}} d\mu = \sum_{k \geq 1} \int_{[0,1]} \chi_{I_k \cap A} d \mu](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmu+X%5E%7B-1%7D%28A%29+%3D+%5Cmu%28A%29+%3D+%5Cint_A+d%5Cmu+%3D+%5Cint_%7B%5B0%2C1%5D%7D+%5Csum_%7Bk+%5Cgeq+1%7D+%7B%5Cchi_%7BI_k+%5Ccap+A%7D%7D+d%5Cmu+%3D+%5Csum_%7Bk+%5Cgeq+1%7D+%5Cint_%7B%5B0%2C1%5D%7D+%5Cchi_%7BI_k+%5Ccap+A%7D+d+%5Cmu+&bg=ffffff&fg=000000&s=-2 "\displaystyle \mu X^{-1}(A) = \mu(A) = \int_A d\mu = \int_{[0,1]} \sum_{k \geq 1} {\chi_{I_k \cap A}} d\mu = \sum_{k \geq 1} \int_{[0,1]} \chi_{I_k \cap A} d \mu")
where the last equality holds by m.c.t ( 
denotes the indicator function). Let 
for 
then note that the restriction of to 
( which we denote by 
) is 1-1, so 
makes sense and its derivative exists so by change of variables we may write,
![\displaystyle \begin{array}{rcl} \mu(S_k) = \int_{S_k} d\mu = \int_{X(S_k)} \left |dX_k^{-1}(X_k(s))/d\mu \right |d\mu(s) \\ = \int_{X([0,1])} \chi_{X(S_k)} \left| \frac{1}{ X_k'(X_k^{-1}(s)) } \right|\,d\mu (s) \end{array}](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cbegin%7Barray%7D%7Brcl%7D++%5Cmu%28S_k%29+%3D+%5Cint_%7BS_k%7D+d%5Cmu+%3D+%5Cint_%7BX%28S_k%29%7D+%5Cleft+%7CdX_k%5E%7B-1%7D%28X_k%28s%29%29%2Fd%5Cmu+%5Cright+%7Cd%5Cmu%28s%29+%5C%5C+%3D+%5Cint_%7BX%28%5B0%2C1%5D%29%7D+%5Cchi_%7BX%28S_k%29%7D+%5Cleft%7C+%5Cfrac%7B1%7D%7B+X_k%27%28X_k%5E%7B-1%7D%28s%29%29+%7D+%5Cright%7C%5C%2Cd%5Cmu+%28s%29+%5Cend%7Barray%7D+&bg=ffffff&fg=000000&s=-2 "\displaystyle \begin{array}{rcl} \mu(S_k) = \int_{S_k} d\mu = \int_{X(S_k)} \left |dX_k^{-1}(X_k(s))/d\mu \right |d\mu(s) \\ = \int_{X([0,1])} \chi_{X(S_k)} \left| \frac{1}{ X_k'(X_k^{-1}(s)) } \right|\,d\mu (s) \end{array}")
So,
![\displaystyle \begin{array}{rcl} \mu X^{-1}(A) = \sum_{k \geq 1} \int_{X[0,1]} \chi_{X(S_k)} \left| \frac{1}{ X'(X^{-1}(s)) } \right|\,d\mu(s)\\ = \int_{X[0,1]} \sum_{k \geq 1}\chi_{X(S_k)} \left| \frac{1}{ X'(X^{-1}(s)) } \right|\,d\mu (s) \end{array}](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cbegin%7Barray%7D%7Brcl%7D++%5Cmu+X%5E%7B-1%7D%28A%29+%3D+%5Csum_%7Bk+%5Cgeq+1%7D+%5Cint_%7BX%5B0%2C1%5D%7D+%5Cchi_%7BX%28S_k%29%7D+%5Cleft%7C+%5Cfrac%7B1%7D%7B+X%27%28X%5E%7B-1%7D%28s%29%29+%7D+%5Cright%7C%5C%2Cd%5Cmu%28s%29%5C%5C+%3D+%5Cint_%7BX%5B0%2C1%5D%7D+%5Csum_%7Bk+%5Cgeq+1%7D%5Cchi_%7BX%28S_k%29%7D+%5Cleft%7C+%5Cfrac%7B1%7D%7B+X%27%28X%5E%7B-1%7D%28s%29%29+%7D+%5Cright%7C%5C%2Cd%5Cmu+%28s%29+%5Cend%7Barray%7D+&bg=ffffff&fg=000000&s=-2 "\displaystyle \begin{array}{rcl} \mu X^{-1}(A) = \sum_{k \geq 1} \int_{X[0,1]} \chi_{X(S_k)} \left| \frac{1}{ X'(X^{-1}(s)) } \right|\,d\mu(s)\\ = \int_{X[0,1]} \sum_{k \geq 1}\chi_{X(S_k)} \left| \frac{1}{ X'(X^{-1}(s)) } \right|\,d\mu (s) \end{array}")
But the density function of is 
as is given to have the uniform distribution, the integrand on the right must be a.e. Thus, for almost all ![{s \in X[0,1]}](http://s0.wp.com/latex.php?latex=%7Bs+%5Cin+X%5B0%2C1%5D%7D&bg=ffffff&fg=000000&s=-2 "{s \in X[0,1]}")
,
Here is a simple example: Consider the random variable as shown in the diagram which has uniform distribution. Notice that for each 
in the range of , that is not in 0 or 1, the sum of the inverse of slopes at each point where assumes the value is .
