In my notes I have the statement:
If $f$ is Lebesgue integrable on $E$, $f \in L(\mathbb{R}^2)$, then for any $y \in \mathbb{R}^d$, $f(x+y) \in L(\mathbb{R}^d) $ and
$$ \int_{\mathbb{R}^d} f(x+y) \ dx = \int_{\mathbb{R}^d} f(x) \ dx $$
I was wondering how this is the case if you add something to $f(x) $ wouldn't you change the integral as well? Consider $f(x) = x^2$. Then $F(x) = x^3/3$. Now consider $f(x+2) = (x+2)^2$ Then its integral is $4x + 2x^2 + x^3/3$.
what does it mean for to be $f \in L(\mathbb{R}^2)$?