How can I prove $$int[F(x+a)-F(x)]\,dx=a$$

How can I prove $$int[F(x+a)-F(x)]\,dx=a$$

where $F(x)$ is the cumulative distribution function?

Proof:

Let $R, S> 0$ be large compared to $a$. Then

$$egin{align*} int_{-R}^{S} left[ F(x+a) - F(x) ight] ; dx &= int_{-R}^{S} F(x+a) ; dx - int_{-R}^{S} F(x); dx \ &= int_{-R+a}^{S+a} F(x) ; dx - int_{-R}^{S} F(x); dx \ &= int_{S}^{S+a} F(x) ; dx - int_{-R}^{-R+a} F(x); dx \ &= int_{0}^{a} F(x+S) ; dx - int_{0}^{a} F(x-R); dx end{align*}$$

Now taking $R, S o infty$, Bounded Convergence Theorem shows that

$$ lim_{S oinfty} int_{0}^{a} F(x+S) ; dx = int_{0}^{a} lim_{S oinfty} F(x+S) ; dx = a$$

and

$$ lim_{R oinfty} int_{0}^{a} F(x-R) ; dx = int_{0}^{a} lim_{R oinfty} F(x-R) ; dx = 0$$

Therefore we have

$$ int_{-infty}^{infty} left[ F(x+a) - F(x) ight] ; dx = a.$$

Proof 2:

We can also prove it using Fubini's theorem for non-negative functions. Let $X$ a random variable of cumulative distribution function $F$, and $(Omega,mathcal F,P)$ the probability space on which $X$ is defined. We have egin{align*} int_{mathbb R}[F(x+a)-F(x)]dx&=int_{mathbb R}int_{Omega}chi_{{(u,v),u<vleq v+a}}(x,X(omega))dP(omega)dx\ &=int_{Omega}int_{mathbb R}chi_{{(u,v),u<vleq v+a}}(x,X(omega))dxdP(omega)\ &=int_{Omega}int_{X(omega)-a}^{X(omega)}dxdP(omega)\ &=int_{Omega}adP(omega)\ &=a. end{align*}