Show $\int _{\mathbb{R}}[F(x+c) - F(x)] dx =0 $ is incorrect Proof $F$ is CDF  (probability distribution function)
$\int _{\mathbb{R}}[F(x+c) - F(x)]  dx $
$= \int _{\mathbb{R}}F(x+c)  dx -\int _{\mathbb{R}} F(x)  dx$  ( (by linearity of the integral)
$=\int _{\mathbb{R}}F(x)  dx -\int _{\mathbb{R}}F(x)  dx $ ( By change of variables theorem )
$=0$
For change of variables we used the known theorem (see below) with $\Omega = \Omega' = \mathbb{R} $ , $T(x) = x + c $ and $ \mu= \lambda$ ( Lebesgue measure)
Question: Why is the above Proof incorrect? How can we show it is not correct analytically?
My thought:
I am not sure how to show it, maybe
because the density does not exist? ( Even assumption of continuity would not imply the existence of the density) ( e.g. Cantor function is a continuous CDF which does not have a density with respect to Lebesgue measure)
Below is the known Theorem that we used 



Why is the above proof wrong ?
In other words why is this step $=\int _{\mathbb{R}}F(x)  dx -\int _{\mathbb{R}}F(x) 
> dx $ ( By change of variables theorem )  Incorrect ?
We use the change of variable theorem (see above) with $\Omega =
 \Omega' = \mathbb{R} $ , $T(x) = x + c $ and $ \mu= \lambda$ (
Lebesgue measure)

 A: For any CDF $F$ we have $\int_{\mathbb R} F(x)dx=\infty$. This is because $F(x) \to 1$ as $ x \to \infty$. If we choose $a$ such that $F(x)>\frac 1  2$ for all $x >a$ then we get $\int_{\mathbb R} F(x)dx \geq \int_a^{\infty} \frac 1 2  dx=\infty$. So splitting the integral into two terms gives you $\infty-\infty$.
For an example where the integral is not $0$ consider $exp(1)$ distribution and take $c=1$ You can easily compute the integral in this case and the answer is not $0$.
Actually the value of the integral is $c$. Let $\mu$ the probability measure corresponding to $F$. Then (by Fubini/Tonelli's Theorem)  $\int_{\mathbb R} [F(x+c)-F(x)]dx=\int_{\mathbb R} \int_{(x,x+c]} d\mu(y)dx=\int \int_{y-c}^{c}dx d\mu (y)=\int c d\mu(y)=c$.
A: Since $F$ is a CDF, $\displaystyle\lim_{x\rightarrow\infty}F(x)=1$. Therefore, the integral $\int_{\mathbb{R}}F(x)\,\mathrm{d}x$ diverges. When you get the expression
$$\int _{\mathbb{R}}F(x+c)\, \mathrm{d}x -\int _{\mathbb{R}} F(x)\, \mathrm{d}x$$
this becomes $\infty-\infty$, which is an indeterminate form, so it cannot be simplified to $0$.
With regard to how one would correctly evaluate this integral, there is a simple way to do it using Fubini's Theorem.
