# Prove $\ \int_E f(x)dx=\int_E f(x+t)dx \$

Let $\ f \$ be integrable on $\ E \$.

Prove $\ \int_E f(x)dx=\int_E f(x+t)dx \$ where $\ E=(-\infty, \infty) \$ and $\ E \$ is measurable

Since $\ f \$ is integrable , we have

$\int_E f(x)dx <\infty \$

Let $E_1,E_2,......,E_N \$ are the disjoint partition (interval) of $E \$ such that

$f(x)=a_i , \ \ if \ \ x\in E_i \$

Let $\ \mu \$ be the measure on E , then $\ \mu(E)=|E| \$

Then,

$\int_E f(x)dx=\sum_{i=0}^{N} a_i |E_i|=\int_E f(t+x)dx \ \\ \\ \Rightarrow \int_E f(x)dx= \int_E f(t+x)dx$

(Proved)

But this concept works if $\ f \$ be a simple function.

I need help to overcome this.

Assume first that $f\geq 0$, then find an increasing sequence $0\leq\varphi_{n}\leq f$ of simple functions such that $\varphi_{n}\uparrow f$ a.e., and we have by Monotone Convergence Theorem that \begin{align*} \int f=\lim_{n}\int\varphi_{n}=\lim_{n}\int\varphi_{n}(\cdot+t)=\int f(\cdot+t). \end{align*} For general functions $f$, split $f=f^{+}-f^{-}$ and tackle separately to $f^{+}$ and $f^{-}$.
• Do you mean $\int _E f(x)dx=\lim_{n} \int_E \phi_n(x)dx=\lim_{n} \int_E \phi(x+t)dx=\int_E f(x+t)dx \$ ? – Why Apr 10 '18 at 19:30
• So we have to show this for $\ f^{+} \$ and $\ f^{-} \$ separately. Then we have to combine the result. Is it > – Why Apr 10 '18 at 19:37